Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 19.9s

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

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

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 (- Ev mu))) 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 + (Ev - mu))) / 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 + (ev - mu))) / 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 + (Ev - mu))) / 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 + (Ev - mu))) / 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(Ev - mu))) / 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 + (Ev - mu))) / 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[(Ev - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ t_2 := \frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + t\_1\\ \mathbf{if}\;EDonor \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;EDonor \leq -8.5 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;EDonor \leq -5.6 \cdot 10^{-173}:\\ \;\;\;\;t\_1 + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq -3.5 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;EDonor \leq 3.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 4.6 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (- EDonor (- (- Ec mu) Vef)) KbT)) 1.0)))
        (t_1 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)))
        (t_2
         (+
          (/ NaChar (+ (exp (/ (+ Vef (+ EAccept (- Ev mu))) KbT)) 1.0))
          t_1)))
   (if (<= EDonor -2.1e+101)
     t_2
     (if (<= EDonor -8.5e-89)
       t_0
       (if (<= EDonor -5.6e-173)
         (+ t_1 (/ NaChar (+ (exp (/ (- (+ EAccept Ev) mu) KbT)) 1.0)))
         (if (<= EDonor -3.5e-193)
           t_0
           (if (<= EDonor 3.6e-179)
             (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
             (if (<= EDonor 4.6e-20) t_0 t_2))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0);
	double t_1 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double t_2 = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + t_1;
	double tmp;
	if (EDonor <= -2.1e+101) {
		tmp = t_2;
	} else if (EDonor <= -8.5e-89) {
		tmp = t_0;
	} else if (EDonor <= -5.6e-173) {
		tmp = t_1 + (NaChar / (exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	} else if (EDonor <= -3.5e-193) {
		tmp = t_0;
	} else if (EDonor <= 3.6e-179) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else if (EDonor <= 4.6e-20) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (exp(((edonor - ((ec - mu) - vef)) / kbt)) + 1.0d0)
    t_1 = ndchar / (exp((edonor / kbt)) + 1.0d0)
    t_2 = (nachar / (exp(((vef + (eaccept + (ev - mu))) / kbt)) + 1.0d0)) + t_1
    if (edonor <= (-2.1d+101)) then
        tmp = t_2
    else if (edonor <= (-8.5d-89)) then
        tmp = t_0
    else if (edonor <= (-5.6d-173)) then
        tmp = t_1 + (nachar / (exp((((eaccept + ev) - mu) / kbt)) + 1.0d0))
    else if (edonor <= (-3.5d-193)) then
        tmp = t_0
    else if (edonor <= 3.6d-179) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else if (edonor <= 4.6d-20) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0);
	double t_1 = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	double t_2 = (NaChar / (Math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + t_1;
	double tmp;
	if (EDonor <= -2.1e+101) {
		tmp = t_2;
	} else if (EDonor <= -8.5e-89) {
		tmp = t_0;
	} else if (EDonor <= -5.6e-173) {
		tmp = t_1 + (NaChar / (Math.exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	} else if (EDonor <= -3.5e-193) {
		tmp = t_0;
	} else if (EDonor <= 3.6e-179) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else if (EDonor <= 4.6e-20) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0)
	t_1 = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	t_2 = (NaChar / (math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + t_1
	tmp = 0
	if EDonor <= -2.1e+101:
		tmp = t_2
	elif EDonor <= -8.5e-89:
		tmp = t_0
	elif EDonor <= -5.6e-173:
		tmp = t_1 + (NaChar / (math.exp((((EAccept + Ev) - mu) / KbT)) + 1.0))
	elif EDonor <= -3.5e-193:
		tmp = t_0
	elif EDonor <= 3.6e-179:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	elif EDonor <= 4.6e-20:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Float64(Float64(Ec - mu) - Vef)) / KbT)) + 1.0))
	t_1 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	t_2 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(EAccept + Float64(Ev - mu))) / KbT)) + 1.0)) + t_1)
	tmp = 0.0
	if (EDonor <= -2.1e+101)
		tmp = t_2;
	elseif (EDonor <= -8.5e-89)
		tmp = t_0;
	elseif (EDonor <= -5.6e-173)
		tmp = Float64(t_1 + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)) + 1.0)));
	elseif (EDonor <= -3.5e-193)
		tmp = t_0;
	elseif (EDonor <= 3.6e-179)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	elseif (EDonor <= 4.6e-20)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0);
	t_1 = NdChar / (exp((EDonor / KbT)) + 1.0);
	t_2 = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + t_1;
	tmp = 0.0;
	if (EDonor <= -2.1e+101)
		tmp = t_2;
	elseif (EDonor <= -8.5e-89)
		tmp = t_0;
	elseif (EDonor <= -5.6e-173)
		tmp = t_1 + (NaChar / (exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	elseif (EDonor <= -3.5e-193)
		tmp = t_0;
	elseif (EDonor <= 3.6e-179)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	elseif (EDonor <= 4.6e-20)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(EDonor - N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / N[(N[Exp[N[(N[(Vef + N[(EAccept + N[(Ev - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[EDonor, -2.1e+101], t$95$2, If[LessEqual[EDonor, -8.5e-89], t$95$0, If[LessEqual[EDonor, -5.6e-173], N[(t$95$1 + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, -3.5e-193], t$95$0, If[LessEqual[EDonor, 3.6e-179], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 4.6e-20], t$95$0, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if EDonor < -2.1e101 or 4.5999999999999998e-20 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 89.0%

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

   7. Simplified 89.0%

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

   if -2.1e101 < EDonor < -8.49999999999999937e-89 or -5.5999999999999998e-173 < EDonor < -3.50000000000000005e-193 or 3.60000000000000007e-179 < EDonor < 4.5999999999999998e-20

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 80.6%

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

   7. Simplified 80.6%

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

   if -8.49999999999999937e-89 < EDonor < -5.5999999999999998e-173

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 94.8%

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

   10. Simplified 94.8%

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

   if -3.50000000000000005e-193 < EDonor < 3.60000000000000007e-179

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 73.5%

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

   7. Simplified 73.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq -8.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq -5.6 \cdot 10^{-173}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq -3.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 3.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 4.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;EDonor \leq -2.2 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;EDonor \leq -9 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;EDonor \leq -1.7 \cdot 10^{-172}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;EDonor \leq -1.05 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;EDonor \leq 2.9 \cdot 10^{-179}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (- EDonor (- (- Ec mu) Vef)) KbT)) 1.0)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept Ev) mu) KbT)) 1.0)))))
   (if (<= EDonor -2.2e+101)
     t_1
     (if (<= EDonor -9e-89)
       t_0
       (if (<= EDonor -1.7e-172)
         t_1
         (if (<= EDonor -1.05e-192)
           t_0
           (if (<= EDonor 2.9e-179)
             (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
             (if (<= EDonor 1.5e-21) 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 = NdChar / (exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0);
	double t_1 = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	double tmp;
	if (EDonor <= -2.2e+101) {
		tmp = t_1;
	} else if (EDonor <= -9e-89) {
		tmp = t_0;
	} else if (EDonor <= -1.7e-172) {
		tmp = t_1;
	} else if (EDonor <= -1.05e-192) {
		tmp = t_0;
	} else if (EDonor <= 2.9e-179) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else if (EDonor <= 1.5e-21) {
		tmp = 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 = ndchar / (exp(((edonor - ((ec - mu) - vef)) / kbt)) + 1.0d0)
    t_1 = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + ev) - mu) / kbt)) + 1.0d0))
    if (edonor <= (-2.2d+101)) then
        tmp = t_1
    else if (edonor <= (-9d-89)) then
        tmp = t_0
    else if (edonor <= (-1.7d-172)) then
        tmp = t_1
    else if (edonor <= (-1.05d-192)) then
        tmp = t_0
    else if (edonor <= 2.9d-179) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else if (edonor <= 1.5d-21) then
        tmp = 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 = NdChar / (Math.exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0);
	double t_1 = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	double tmp;
	if (EDonor <= -2.2e+101) {
		tmp = t_1;
	} else if (EDonor <= -9e-89) {
		tmp = t_0;
	} else if (EDonor <= -1.7e-172) {
		tmp = t_1;
	} else if (EDonor <= -1.05e-192) {
		tmp = t_0;
	} else if (EDonor <= 2.9e-179) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else if (EDonor <= 1.5e-21) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0)
	t_1 = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + Ev) - mu) / KbT)) + 1.0))
	tmp = 0
	if EDonor <= -2.2e+101:
		tmp = t_1
	elif EDonor <= -9e-89:
		tmp = t_0
	elif EDonor <= -1.7e-172:
		tmp = t_1
	elif EDonor <= -1.05e-192:
		tmp = t_0
	elif EDonor <= 2.9e-179:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	elif EDonor <= 1.5e-21:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Float64(Float64(Ec - mu) - Vef)) / KbT)) + 1.0))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (EDonor <= -2.2e+101)
		tmp = t_1;
	elseif (EDonor <= -9e-89)
		tmp = t_0;
	elseif (EDonor <= -1.7e-172)
		tmp = t_1;
	elseif (EDonor <= -1.05e-192)
		tmp = t_0;
	elseif (EDonor <= 2.9e-179)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	elseif (EDonor <= 1.5e-21)
		tmp = 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 = NdChar / (exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0);
	t_1 = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (EDonor <= -2.2e+101)
		tmp = t_1;
	elseif (EDonor <= -9e-89)
		tmp = t_0;
	elseif (EDonor <= -1.7e-172)
		tmp = t_1;
	elseif (EDonor <= -1.05e-192)
		tmp = t_0;
	elseif (EDonor <= 2.9e-179)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	elseif (EDonor <= 1.5e-21)
		tmp = 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[(NdChar / N[(N[Exp[N[(N[(EDonor - N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EDonor, -2.2e+101], t$95$1, If[LessEqual[EDonor, -9e-89], t$95$0, If[LessEqual[EDonor, -1.7e-172], t$95$1, If[LessEqual[EDonor, -1.05e-192], t$95$0, If[LessEqual[EDonor, 2.9e-179], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 1.5e-21], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if EDonor < -2.2000000000000001e101 or -8.9999999999999998e-89 < EDonor < -1.6999999999999999e-172 or 1.49999999999999996e-21 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 87.4%

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

   10. Simplified 87.4%

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

   if -2.2000000000000001e101 < EDonor < -8.9999999999999998e-89 or -1.6999999999999999e-172 < EDonor < -1.04999999999999997e-192 or 2.8999999999999999e-179 < EDonor < 1.49999999999999996e-21

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 80.6%

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

   7. Simplified 80.6%

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

   if -1.04999999999999997e-192 < EDonor < 2.8999999999999999e-179

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 73.5%

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

   7. Simplified 73.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -2.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq -9 \cdot 10^{-89}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq -1.7 \cdot 10^{-172}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq -1.05 \cdot 10^{-192}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 2.9 \cdot 10^{-179}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ t_2 := \frac{NdChar}{2} + t\_0\\ \mathbf{if}\;KbT \leq -22000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;KbT \leq -7.5 \cdot 10^{-245}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq 4 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.15 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ Ev KbT)) 1.0)))
        (t_1 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)))
        (t_2 (+ (/ NdChar 2.0) t_0)))
   (if (<= KbT -22000000000000.0)
     t_2
     (if (<= KbT -7.5e-245)
       t_1
       (if (<= KbT 4e+32) t_0 (if (<= KbT 2.15e+138) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((Ev / KbT)) + 1.0);
	double t_1 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double t_2 = (NdChar / 2.0) + t_0;
	double tmp;
	if (KbT <= -22000000000000.0) {
		tmp = t_2;
	} else if (KbT <= -7.5e-245) {
		tmp = t_1;
	} else if (KbT <= 4e+32) {
		tmp = t_0;
	} else if (KbT <= 2.15e+138) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (exp((ev / kbt)) + 1.0d0)
    t_1 = ndchar / (exp((edonor / kbt)) + 1.0d0)
    t_2 = (ndchar / 2.0d0) + t_0
    if (kbt <= (-22000000000000.0d0)) then
        tmp = t_2
    else if (kbt <= (-7.5d-245)) then
        tmp = t_1
    else if (kbt <= 4d+32) then
        tmp = t_0
    else if (kbt <= 2.15d+138) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	double t_1 = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	double t_2 = (NdChar / 2.0) + t_0;
	double tmp;
	if (KbT <= -22000000000000.0) {
		tmp = t_2;
	} else if (KbT <= -7.5e-245) {
		tmp = t_1;
	} else if (KbT <= 4e+32) {
		tmp = t_0;
	} else if (KbT <= 2.15e+138) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((Ev / KbT)) + 1.0)
	t_1 = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	t_2 = (NdChar / 2.0) + t_0
	tmp = 0
	if KbT <= -22000000000000.0:
		tmp = t_2
	elif KbT <= -7.5e-245:
		tmp = t_1
	elif KbT <= 4e+32:
		tmp = t_0
	elif KbT <= 2.15e+138:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0))
	t_1 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	t_2 = Float64(Float64(NdChar / 2.0) + t_0)
	tmp = 0.0
	if (KbT <= -22000000000000.0)
		tmp = t_2;
	elseif (KbT <= -7.5e-245)
		tmp = t_1;
	elseif (KbT <= 4e+32)
		tmp = t_0;
	elseif (KbT <= 2.15e+138)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((Ev / KbT)) + 1.0);
	t_1 = NdChar / (exp((EDonor / KbT)) + 1.0);
	t_2 = (NdChar / 2.0) + t_0;
	tmp = 0.0;
	if (KbT <= -22000000000000.0)
		tmp = t_2;
	elseif (KbT <= -7.5e-245)
		tmp = t_1;
	elseif (KbT <= 4e+32)
		tmp = t_0;
	elseif (KbT <= 2.15e+138)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / 2.0), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[KbT, -22000000000000.0], t$95$2, If[LessEqual[KbT, -7.5e-245], t$95$1, If[LessEqual[KbT, 4e+32], t$95$0, If[LessEqual[KbT, 2.15e+138], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -2.2e13 or 2.1499999999999999e138 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 75.9%

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

   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 66.9%

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

   10. Simplified 66.9%

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

   if -2.2e13 < KbT < -7.5000000000000003e-245 or 4.00000000000000021e32 < KbT < 2.1499999999999999e138

   1. Initial program 100.0%

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

   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 EDonor around inf

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

      1. /-lowering-/.f64 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 51.9%

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

   10. Simplified 51.9%

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

   if -7.5000000000000003e-245 < KbT < 4.00000000000000021e32

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 63.1%

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

   7. Simplified 63.1%

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

   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 36.6%

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

   10. Simplified 36.6%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -22000000000000:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq -7.5 \cdot 10^{-245}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 4 \cdot 10^{+32}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 2.15 \cdot 10^{+138}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ t_1 := \frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -1.8 \cdot 10^{+261}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq -1.65 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 1.75 \cdot 10^{+135}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (- EDonor (- (- Ec mu) Vef)) KbT)) 1.0)))
        (t_1 (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))))
   (if (<= NaChar -1.8e+261)
     t_0
     (if (<= NaChar -1.65e+131) t_1 (if (<= NaChar 1.75e+135) 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 = NdChar / (exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0);
	double t_1 = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.8e+261) {
		tmp = t_0;
	} else if (NaChar <= -1.65e+131) {
		tmp = t_1;
	} else if (NaChar <= 1.75e+135) {
		tmp = 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 = ndchar / (exp(((edonor - ((ec - mu) - vef)) / kbt)) + 1.0d0)
    t_1 = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    if (nachar <= (-1.8d+261)) then
        tmp = t_0
    else if (nachar <= (-1.65d+131)) then
        tmp = t_1
    else if (nachar <= 1.75d+135) then
        tmp = 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 = NdChar / (Math.exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0);
	double t_1 = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.8e+261) {
		tmp = t_0;
	} else if (NaChar <= -1.65e+131) {
		tmp = t_1;
	} else if (NaChar <= 1.75e+135) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0)
	t_1 = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -1.8e+261:
		tmp = t_0
	elif NaChar <= -1.65e+131:
		tmp = t_1
	elif NaChar <= 1.75e+135:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Float64(Float64(Ec - mu) - Vef)) / KbT)) + 1.0))
	t_1 = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -1.8e+261)
		tmp = t_0;
	elseif (NaChar <= -1.65e+131)
		tmp = t_1;
	elseif (NaChar <= 1.75e+135)
		tmp = 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 = NdChar / (exp(((EDonor - ((Ec - mu) - Vef)) / KbT)) + 1.0);
	t_1 = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -1.8e+261)
		tmp = t_0;
	elseif (NaChar <= -1.65e+131)
		tmp = t_1;
	elseif (NaChar <= 1.75e+135)
		tmp = 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[(NdChar / N[(N[Exp[N[(N[(EDonor - N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.8e+261], t$95$0, If[LessEqual[NaChar, -1.65e+131], t$95$1, If[LessEqual[NaChar, 1.75e+135], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.80000000000000009e261 or -1.6499999999999999e131 < NaChar < 1.7500000000000001e135

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 74.2%

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

   7. Simplified 74.2%

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

   if -1.80000000000000009e261 < NaChar < -1.6499999999999999e131 or 1.7500000000000001e135 < 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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.8 \cdot 10^{+261}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -1.65 \cdot 10^{+131}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 1.75 \cdot 10^{+135}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - \left(\left(Ec - mu\right) - Vef\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ t_1 := \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -4.1 \cdot 10^{+177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq 5.8 \cdot 10^{-304}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 2.2 \cdot 10^{-187}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)))
        (t_1 (/ NaChar (+ (exp (/ Vef KbT)) 1.0))))
   (if (<= Vef -4.1e+177)
     t_1
     (if (<= Vef 5.8e-304)
       t_0
       (if (<= Vef 2.2e-187)
         (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
         (if (<= Vef 3.8e+145) 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 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double t_1 = NaChar / (exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -4.1e+177) {
		tmp = t_1;
	} else if (Vef <= 5.8e-304) {
		tmp = t_0;
	} else if (Vef <= 2.2e-187) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (Vef <= 3.8e+145) {
		tmp = 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 = ndchar / (exp((edonor / kbt)) + 1.0d0)
    t_1 = nachar / (exp((vef / kbt)) + 1.0d0)
    if (vef <= (-4.1d+177)) then
        tmp = t_1
    else if (vef <= 5.8d-304) then
        tmp = t_0
    else if (vef <= 2.2d-187) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else if (vef <= 3.8d+145) then
        tmp = 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 = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	double t_1 = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -4.1e+177) {
		tmp = t_1;
	} else if (Vef <= 5.8e-304) {
		tmp = t_0;
	} else if (Vef <= 2.2e-187) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else if (Vef <= 3.8e+145) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	t_1 = NaChar / (math.exp((Vef / KbT)) + 1.0)
	tmp = 0
	if Vef <= -4.1e+177:
		tmp = t_1
	elif Vef <= 5.8e-304:
		tmp = t_0
	elif Vef <= 2.2e-187:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	elif Vef <= 3.8e+145:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	t_1 = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0))
	tmp = 0.0
	if (Vef <= -4.1e+177)
		tmp = t_1;
	elseif (Vef <= 5.8e-304)
		tmp = t_0;
	elseif (Vef <= 2.2e-187)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (Vef <= 3.8e+145)
		tmp = 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 = NdChar / (exp((EDonor / KbT)) + 1.0);
	t_1 = NaChar / (exp((Vef / KbT)) + 1.0);
	tmp = 0.0;
	if (Vef <= -4.1e+177)
		tmp = t_1;
	elseif (Vef <= 5.8e-304)
		tmp = t_0;
	elseif (Vef <= 2.2e-187)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	elseif (Vef <= 3.8e+145)
		tmp = 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[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -4.1e+177], t$95$1, If[LessEqual[Vef, 5.8e-304], t$95$0, If[LessEqual[Vef, 2.2e-187], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 3.8e+145], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -4.10000000000000014e177 or 3.80000000000000012e145 < 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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 72.9%

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

   7. Simplified 72.9%

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

   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 69.9%

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

   10. Simplified 69.9%

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

   if -4.10000000000000014e177 < Vef < 5.8e-304 or 2.20000000000000008e-187 < Vef < 3.80000000000000012e145

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 75.8%

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

   7. Simplified 75.8%

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

   8. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 48.3%

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

   10. Simplified 48.3%

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

   if 5.8e-304 < Vef < 2.20000000000000008e-187

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 70.1%

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

   7. Simplified 70.1%

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

   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 52.0%

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

   10. Simplified 52.0%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -4.1 \cdot 10^{+177}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 5.8 \cdot 10^{-304}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 2.2 \cdot 10^{-187}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.05 \cdot 10^{+155}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -1.05e+155)
   (+ (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (/ NaChar 2.0))
   (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) 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 (NdChar <= -1.05e+155) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	} else {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / 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 (ndchar <= (-1.05d+155)) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar / 2.0d0)
    else
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / 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 (NdChar <= -1.05e+155) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	} else {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -1.05e+155:
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0)
	else:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -1.05e+155)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar / 2.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / 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 (NdChar <= -1.05e+155)
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	else
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -1.05e+155], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.05e155

   1. Initial program 100.0%

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

   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 EDonor around inf

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

      1. /-lowering-/.f64 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in KbT around inf

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

   10. Simplified 58.7%

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

   if -1.05e155 < NdChar

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 63.0%

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

   7. Simplified 63.0%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.05 \cdot 10^{+155}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -7.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;KbT \leq 2.9 \cdot 10^{+103}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -7.2e-14)
   (+ (/ NdChar 2.0) (/ NaChar (+ 2.0 (/ Ev KbT))))
   (if (<= KbT 2.9e+103)
     (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
     (+
      (/ NdChar 2.0)
      (/
       NaChar
       (-
        (+ 2.0 (+ (/ Vef KbT) (+ (/ Ev KbT) (/ EAccept KbT))))
        (/ mu KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -7.2e-14) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (KbT <= 2.9e+103) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-7.2d-14)) then
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 + (ev / kbt)))
    else if (kbt <= 2.9d+103) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = (ndchar / 2.0d0) + (nachar / ((2.0d0 + ((vef / kbt) + ((ev / kbt) + (eaccept / kbt)))) - (mu / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -7.2e-14) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (KbT <= 2.9e+103) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -7.2e-14:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)))
	elif KbT <= 2.9e+103:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -7.2e-14)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))));
	elseif (KbT <= 2.9e+103)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(Vef / KbT) + Float64(Float64(Ev / KbT) + Float64(EAccept / KbT)))) - Float64(mu / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -7.2e-14)
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	elseif (KbT <= 2.9e+103)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -7.2e-14], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.9e+103], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(N[(2.0 + N[(N[(Vef / KbT), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -7.1999999999999996e-14

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 71.7%

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

   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 62.9%

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

   10. Simplified 62.9%

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

   11. Taylor expanded in Ev around 0

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

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

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

       2. /-lowering-/.f64 53.2%

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

   13. Simplified 53.2%

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

   if -7.1999999999999996e-14 < KbT < 2.8999999999999998e103

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 61.1%

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

   7. Simplified 61.1%

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

   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 37.5%

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

   10. Simplified 37.5%

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

   if 2.8999999999999998e103 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 71.5%

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

   8. Taylor expanded in KbT around inf

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

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

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

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

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

      3. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right) + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{EAccept}{KbT}\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 63.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \mathsf{/.f64}\left(mu, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 63.6%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -7.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;KbT \leq 2.9 \cdot 10^{+103}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.42 \cdot 10^{+78}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4e+25)
   (+ (/ NdChar 2.0) (/ NaChar (+ 2.0 (/ Ev KbT))))
   (if (<= KbT 1.42e+78)
     (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
     (+
      (/ NdChar 2.0)
      (/
       NaChar
       (-
        (+ 2.0 (+ (/ Vef KbT) (+ (/ Ev KbT) (/ EAccept KbT))))
        (/ mu KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4e+25) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (KbT <= 1.42e+78) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-4d+25)) then
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 + (ev / kbt)))
    else if (kbt <= 1.42d+78) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else
        tmp = (ndchar / 2.0d0) + (nachar / ((2.0d0 + ((vef / kbt) + ((ev / kbt) + (eaccept / kbt)))) - (mu / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4e+25) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (KbT <= 1.42e+78) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -4e+25:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)))
	elif KbT <= 1.42e+78:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	else:
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4e+25)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))));
	elseif (KbT <= 1.42e+78)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	else
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(Vef / KbT) + Float64(Float64(Ev / KbT) + Float64(EAccept / KbT)))) - Float64(mu / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -4e+25)
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	elseif (KbT <= 1.42e+78)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	else
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4e+25], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.42e+78], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(N[(2.0 + N[(N[(Vef / KbT), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.00000000000000036e25

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 74.3%

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

   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 66.5%

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

   10. Simplified 66.5%

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

   11. Taylor expanded in Ev around 0

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

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

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

       2. /-lowering-/.f64 57.5%

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

   13. Simplified 57.5%

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

   if -4.00000000000000036e25 < KbT < 1.42e78

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 59.8%

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

   7. Simplified 59.8%

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

   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 34.2%

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

   10. Simplified 34.2%

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

   if 1.42e78 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 64.6%

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

   8. Taylor expanded in KbT around inf

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

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

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

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

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

      3. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right) + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{EAccept}{KbT}\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 57.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \mathsf{/.f64}\left(mu, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 57.5%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.42 \cdot 10^{+78}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -2.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.08 \cdot 10^{+102}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -2.3e+21)
   (+ (/ NdChar 2.0) (/ NaChar (+ 2.0 (/ Ev KbT))))
   (if (<= KbT 1.08e+102)
     (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
     (+
      (/ NdChar 2.0)
      (/
       NaChar
       (-
        (+ 2.0 (+ (/ Vef KbT) (+ (/ Ev KbT) (/ EAccept KbT))))
        (/ mu KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -2.3e+21) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (KbT <= 1.08e+102) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-2.3d+21)) then
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 + (ev / kbt)))
    else if (kbt <= 1.08d+102) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else
        tmp = (ndchar / 2.0d0) + (nachar / ((2.0d0 + ((vef / kbt) + ((ev / kbt) + (eaccept / kbt)))) - (mu / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -2.3e+21) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (KbT <= 1.08e+102) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -2.3e+21:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)))
	elif KbT <= 1.08e+102:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	else:
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -2.3e+21)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))));
	elseif (KbT <= 1.08e+102)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(Vef / KbT) + Float64(Float64(Ev / KbT) + Float64(EAccept / KbT)))) - Float64(mu / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -2.3e+21)
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	elseif (KbT <= 1.08e+102)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	else
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -2.3e+21], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.08e+102], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(N[(2.0 + N[(N[(Vef / KbT), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -2.3e21

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 73.8%

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

   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 64.5%

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

   10. Simplified 64.5%

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

   11. Taylor expanded in Ev around 0

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

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

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

       2. /-lowering-/.f64 55.6%

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

   13. Simplified 55.6%

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

   if -2.3e21 < KbT < 1.08000000000000002e102

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 61.1%

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

   7. Simplified 61.1%

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

   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 30.9%

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

   10. Simplified 30.9%

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

   if 1.08000000000000002e102 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 71.5%

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

   8. Taylor expanded in KbT around inf

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

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

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

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

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

      3. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right) + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{EAccept}{KbT}\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 63.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \mathsf{/.f64}\left(mu, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 63.6%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.08 \cdot 10^{+102}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(EAccept + \left(Vef + Ev\right)\right) - mu\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;KbT \leq -4.7 \cdot 10^{+64}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{t\_0 + \frac{0.16666666666666666 \cdot \frac{t\_0 \cdot t\_1}{KbT} + t\_1 \cdot 0.5}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (+ EAccept (+ Vef Ev)) mu)) (t_1 (* t_0 t_0)))
   (if (<= KbT -4.7e+64)
     (+ (/ NdChar 2.0) (/ NaChar (+ 2.0 (/ Ev KbT))))
     (if (<= KbT 1.35e+104)
       (/
        NaChar
        (+
         2.0
         (/
          (+
           t_0
           (/ (+ (* 0.16666666666666666 (/ (* t_0 t_1) KbT)) (* t_1 0.5)) KbT))
          KbT)))
       (+
        (/ NdChar 2.0)
        (/
         NaChar
         (-
          (+ 2.0 (+ (/ Vef KbT) (+ (/ Ev KbT) (/ EAccept KbT))))
          (/ mu KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (EAccept + (Vef + Ev)) - mu;
	double t_1 = t_0 * t_0;
	double tmp;
	if (KbT <= -4.7e+64) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (KbT <= 1.35e+104) {
		tmp = NaChar / (2.0 + ((t_0 + (((0.16666666666666666 * ((t_0 * t_1) / KbT)) + (t_1 * 0.5)) / KbT)) / KbT));
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (eaccept + (vef + ev)) - mu
    t_1 = t_0 * t_0
    if (kbt <= (-4.7d+64)) then
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 + (ev / kbt)))
    else if (kbt <= 1.35d+104) then
        tmp = nachar / (2.0d0 + ((t_0 + (((0.16666666666666666d0 * ((t_0 * t_1) / kbt)) + (t_1 * 0.5d0)) / kbt)) / kbt))
    else
        tmp = (ndchar / 2.0d0) + (nachar / ((2.0d0 + ((vef / kbt) + ((ev / kbt) + (eaccept / kbt)))) - (mu / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (EAccept + (Vef + Ev)) - mu;
	double t_1 = t_0 * t_0;
	double tmp;
	if (KbT <= -4.7e+64) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (KbT <= 1.35e+104) {
		tmp = NaChar / (2.0 + ((t_0 + (((0.16666666666666666 * ((t_0 * t_1) / KbT)) + (t_1 * 0.5)) / KbT)) / KbT));
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (EAccept + (Vef + Ev)) - mu
	t_1 = t_0 * t_0
	tmp = 0
	if KbT <= -4.7e+64:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)))
	elif KbT <= 1.35e+104:
		tmp = NaChar / (2.0 + ((t_0 + (((0.16666666666666666 * ((t_0 * t_1) / KbT)) + (t_1 * 0.5)) / KbT)) / KbT))
	else:
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(EAccept + Float64(Vef + Ev)) - mu)
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (KbT <= -4.7e+64)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))));
	elseif (KbT <= 1.35e+104)
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(t_0 + Float64(Float64(Float64(0.16666666666666666 * Float64(Float64(t_0 * t_1) / KbT)) + Float64(t_1 * 0.5)) / KbT)) / KbT)));
	else
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(Vef / KbT) + Float64(Float64(Ev / KbT) + Float64(EAccept / KbT)))) - Float64(mu / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (EAccept + (Vef + Ev)) - mu;
	t_1 = t_0 * t_0;
	tmp = 0.0;
	if (KbT <= -4.7e+64)
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	elseif (KbT <= 1.35e+104)
		tmp = NaChar / (2.0 + ((t_0 + (((0.16666666666666666 * ((t_0 * t_1) / KbT)) + (t_1 * 0.5)) / KbT)) / KbT));
	else
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[KbT, -4.7e+64], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.35e+104], N[(NaChar / N[(2.0 + N[(N[(t$95$0 + N[(N[(N[(0.16666666666666666 * N[(N[(t$95$0 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(N[(2.0 + N[(N[(Vef / KbT), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.70000000000000029e64

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 79.1%

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

   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 71.9%

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

   10. Simplified 71.9%

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

   11. Taylor expanded in Ev around 0

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

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

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

       2. /-lowering-/.f64 63.3%

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

   13. Simplified 63.3%

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

   if -4.70000000000000029e64 < KbT < 1.34999999999999992e104

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 60.3%

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

   7. Simplified 60.3%

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

   8. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right) \]

   10. Simplified 29.6%

       \[\leadsto \frac{NaChar}{\color{blue}{2 + \left(-\frac{-1 \cdot \left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{0.16666666666666666 \cdot \frac{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) \cdot \left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right)}{KbT} + 0.5 \cdot \left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right)}{KbT}\right)}{KbT}\right)}} \]

   if 1.34999999999999992e104 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 71.5%

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

   8. Taylor expanded in KbT around inf

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

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

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

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

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

      3. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right) + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{EAccept}{KbT}\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 63.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \mathsf{/.f64}\left(mu, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 63.6%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.7 \cdot 10^{+64}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) + \frac{0.16666666666666666 \cdot \frac{\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) \cdot \left(\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) \cdot \left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right)\right)}{KbT} + \left(\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) \cdot \left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right)\right) \cdot 0.5}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EAccept + \left(Vef + Ev\right)\\ t_1 := t\_0 - mu\\ \mathbf{if}\;KbT \leq -3.6 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 7.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{t\_1 + -0.5 \cdot \frac{t\_1 \cdot \left(mu - t\_0\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EAccept (+ Vef Ev))) (t_1 (- t_0 mu)))
   (if (<= KbT -3.6e-33)
     (* 0.5 (+ NdChar NaChar))
     (if (<= KbT 7.1e+103)
       (/ NaChar (+ 2.0 (/ (+ t_1 (* -0.5 (/ (* t_1 (- mu t_0)) KbT))) KbT)))
       (+
        (/ NdChar 2.0)
        (/
         NaChar
         (-
          (+ 2.0 (+ (/ Vef KbT) (+ (/ Ev KbT) (/ EAccept KbT))))
          (/ mu KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EAccept + (Vef + Ev);
	double t_1 = t_0 - mu;
	double tmp;
	if (KbT <= -3.6e-33) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 7.1e+103) {
		tmp = NaChar / (2.0 + ((t_1 + (-0.5 * ((t_1 * (mu - t_0)) / KbT))) / KbT));
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = eaccept + (vef + ev)
    t_1 = t_0 - mu
    if (kbt <= (-3.6d-33)) then
        tmp = 0.5d0 * (ndchar + nachar)
    else if (kbt <= 7.1d+103) then
        tmp = nachar / (2.0d0 + ((t_1 + ((-0.5d0) * ((t_1 * (mu - t_0)) / kbt))) / kbt))
    else
        tmp = (ndchar / 2.0d0) + (nachar / ((2.0d0 + ((vef / kbt) + ((ev / kbt) + (eaccept / kbt)))) - (mu / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EAccept + (Vef + Ev);
	double t_1 = t_0 - mu;
	double tmp;
	if (KbT <= -3.6e-33) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 7.1e+103) {
		tmp = NaChar / (2.0 + ((t_1 + (-0.5 * ((t_1 * (mu - t_0)) / KbT))) / KbT));
	} else {
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = EAccept + (Vef + Ev)
	t_1 = t_0 - mu
	tmp = 0
	if KbT <= -3.6e-33:
		tmp = 0.5 * (NdChar + NaChar)
	elif KbT <= 7.1e+103:
		tmp = NaChar / (2.0 + ((t_1 + (-0.5 * ((t_1 * (mu - t_0)) / KbT))) / KbT))
	else:
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EAccept + Float64(Vef + Ev))
	t_1 = Float64(t_0 - mu)
	tmp = 0.0
	if (KbT <= -3.6e-33)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	elseif (KbT <= 7.1e+103)
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(t_1 + Float64(-0.5 * Float64(Float64(t_1 * Float64(mu - t_0)) / KbT))) / KbT)));
	else
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(Vef / KbT) + Float64(Float64(Ev / KbT) + Float64(EAccept / KbT)))) - Float64(mu / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = EAccept + (Vef + Ev);
	t_1 = t_0 - mu;
	tmp = 0.0;
	if (KbT <= -3.6e-33)
		tmp = 0.5 * (NdChar + NaChar);
	elseif (KbT <= 7.1e+103)
		tmp = NaChar / (2.0 + ((t_1 + (-0.5 * ((t_1 * (mu - t_0)) / KbT))) / KbT));
	else
		tmp = (NdChar / 2.0) + (NaChar / ((2.0 + ((Vef / KbT) + ((Ev / KbT) + (EAccept / KbT)))) - (mu / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - mu), $MachinePrecision]}, If[LessEqual[KbT, -3.6e-33], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 7.1e+103], N[(NaChar / N[(2.0 + N[(N[(t$95$1 + N[(-0.5 * N[(N[(t$95$1 * N[(mu - t$95$0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(N[(2.0 + N[(N[(Vef / KbT), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -3.60000000000000034e-33

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 52.1%

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

   7. Simplified 52.1%

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

   if -3.60000000000000034e-33 < KbT < 7.1000000000000002e103

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in KbT around -inf

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

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

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

      2. mul-1-neg N/A

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

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

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

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

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

   10. Simplified 28.8%

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

   if 7.1000000000000002e103 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 71.5%

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

   8. Taylor expanded in KbT around inf

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

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

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

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

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

      3. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right) + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{EAccept}{KbT} + \frac{Ev}{KbT}\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{EAccept}{KbT}\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \left(\frac{Ev}{KbT}\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 63.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \mathsf{/.f64}\left(mu, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 63.6%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.6 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 7.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) + -0.5 \cdot \frac{\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) \cdot \left(mu - \left(EAccept + \left(Vef + Ev\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(2 + \left(\frac{Vef}{KbT} + \left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{if}\;KbT \leq -9200000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 9 \cdot 10^{+135}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{-1 + \frac{NaChar - \frac{NaChar \cdot NaChar}{NdChar}}{NdChar}}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NdChar 2.0) (/ NaChar (+ 2.0 (/ Ev KbT))))))
   (if (<= KbT -9200000000000.0)
     t_0
     (if (<= KbT 9e+135)
       (*
        0.5
        (/
         -1.0
         (/
          (+ -1.0 (/ (- NaChar (/ (* NaChar NaChar) NdChar)) NdChar))
          NdChar)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	double tmp;
	if (KbT <= -9200000000000.0) {
		tmp = t_0;
	} else if (KbT <= 9e+135) {
		tmp = 0.5 * (-1.0 / ((-1.0 + ((NaChar - ((NaChar * NaChar) / NdChar)) / NdChar)) / NdChar));
	} 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 / 2.0d0) + (nachar / (2.0d0 + (ev / kbt)))
    if (kbt <= (-9200000000000.0d0)) then
        tmp = t_0
    else if (kbt <= 9d+135) then
        tmp = 0.5d0 * ((-1.0d0) / (((-1.0d0) + ((nachar - ((nachar * nachar) / ndchar)) / ndchar)) / ndchar))
    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 / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	double tmp;
	if (KbT <= -9200000000000.0) {
		tmp = t_0;
	} else if (KbT <= 9e+135) {
		tmp = 0.5 * (-1.0 / ((-1.0 + ((NaChar - ((NaChar * NaChar) / NdChar)) / NdChar)) / NdChar));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / 2.0) + (NaChar / (2.0 + (Ev / KbT)))
	tmp = 0
	if KbT <= -9200000000000.0:
		tmp = t_0
	elif KbT <= 9e+135:
		tmp = 0.5 * (-1.0 / ((-1.0 + ((NaChar - ((NaChar * NaChar) / NdChar)) / NdChar)) / NdChar))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))))
	tmp = 0.0
	if (KbT <= -9200000000000.0)
		tmp = t_0;
	elseif (KbT <= 9e+135)
		tmp = Float64(0.5 * Float64(-1.0 / Float64(Float64(-1.0 + Float64(Float64(NaChar - Float64(Float64(NaChar * NaChar) / NdChar)) / NdChar)) / NdChar)));
	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 / 2.0) + (NaChar / (2.0 + (Ev / KbT)));
	tmp = 0.0;
	if (KbT <= -9200000000000.0)
		tmp = t_0;
	elseif (KbT <= 9e+135)
		tmp = 0.5 * (-1.0 / ((-1.0 + ((NaChar - ((NaChar * NaChar) / NdChar)) / NdChar)) / NdChar));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -9200000000000.0], t$95$0, If[LessEqual[KbT, 9e+135], N[(0.5 * N[(-1.0 / N[(N[(-1.0 + N[(N[(NaChar - N[(N[(NaChar * NaChar), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -9.2e12 or 9.00000000000000014e135 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 75.3%

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

   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 66.4%

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

   10. Simplified 66.4%

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

   11. Taylor expanded in Ev around 0

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

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

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

       2. /-lowering-/.f64 60.0%

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

   13. Simplified 60.0%

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

   if -9.2e12 < KbT < 9.00000000000000014e135

   1. Initial program 100.0%

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

   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 12.2%

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

   7. Simplified 12.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   8. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{NaChar}^{3} + {NdChar}^{3}}{\color{blue}{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{\color{blue}{\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \color{blue}{\left({NaChar}^{3} + {NdChar}^{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(NaChar \cdot NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left(\color{blue}{{NaChar}^{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left({\color{blue}{NaChar}}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \left({NaChar}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left({NaChar}^{3}\right), \color{blue}{\left({NdChar}^{3}\right)}\right)\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left(NaChar \cdot \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{*.f64}\left(NaChar, NaChar\right)\right), \left({NdChar}^{3}\right)\right)\right)\right)\right) \]

      14. cube-mult N/A

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

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

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

      16. *-lowering-*.f64 6.0%

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

   9. Applied egg-rr 6.0%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NaChar \cdot NaChar + NdChar \cdot \left(NdChar - NaChar\right)}{NaChar \cdot \left(NaChar \cdot NaChar\right) + NdChar \cdot \left(NdChar \cdot NdChar\right)}}} \]

   10. Taylor expanded in NdChar around -inf

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

       1. associate-*r/ N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. distribute-lft-in N/A

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

       5. associate-*r/ N/A

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

       6. distribute-lft-in N/A

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

       7. neg-mul-1 N/A

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

       8. mul-1-neg N/A

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

       9. remove-double-neg N/A

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

       10. metadata-eval N/A

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

       11. +-commutative N/A

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

       12. /-lowering-/.f64 N/A

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

   12. Simplified 21.6%

       \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 - \frac{NaChar - \frac{NaChar \cdot NaChar}{NdChar}}{NdChar}}{NdChar}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -9200000000000:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;KbT \leq 9 \cdot 10^{+135}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{-1 + \frac{NaChar - \frac{NaChar \cdot NaChar}{NdChar}}{NdChar}}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.0% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -1.55 \cdot 10^{+208}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{NaChar \cdot \left(NdChar - NaChar\right)}{NdChar \cdot \left(NdChar \cdot NdChar\right)}}\\ \mathbf{elif}\;Vef \leq 1.2 \cdot 10^{+268}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{NdChar}{NaChar \cdot NaChar} + \frac{-1}{NaChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Vef -1.55e+208)
   (*
    0.5
    (/ -1.0 (/ (* NaChar (- NdChar NaChar)) (* NdChar (* NdChar NdChar)))))
   (if (<= Vef 1.2e+268)
     (* 0.5 (+ NdChar NaChar))
     (* 0.5 (/ -1.0 (+ (/ NdChar (* NaChar NaChar)) (/ -1.0 NaChar)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Vef <= -1.55e+208) {
		tmp = 0.5 * (-1.0 / ((NaChar * (NdChar - NaChar)) / (NdChar * (NdChar * NdChar))));
	} else if (Vef <= 1.2e+268) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * (-1.0 / ((NdChar / (NaChar * NaChar)) + (-1.0 / NaChar)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (vef <= (-1.55d+208)) then
        tmp = 0.5d0 * ((-1.0d0) / ((nachar * (ndchar - nachar)) / (ndchar * (ndchar * ndchar))))
    else if (vef <= 1.2d+268) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = 0.5d0 * ((-1.0d0) / ((ndchar / (nachar * nachar)) + ((-1.0d0) / nachar)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Vef <= -1.55e+208) {
		tmp = 0.5 * (-1.0 / ((NaChar * (NdChar - NaChar)) / (NdChar * (NdChar * NdChar))));
	} else if (Vef <= 1.2e+268) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * (-1.0 / ((NdChar / (NaChar * NaChar)) + (-1.0 / NaChar)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Vef <= -1.55e+208:
		tmp = 0.5 * (-1.0 / ((NaChar * (NdChar - NaChar)) / (NdChar * (NdChar * NdChar))))
	elif Vef <= 1.2e+268:
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = 0.5 * (-1.0 / ((NdChar / (NaChar * NaChar)) + (-1.0 / NaChar)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Vef <= -1.55e+208)
		tmp = Float64(0.5 * Float64(-1.0 / Float64(Float64(NaChar * Float64(NdChar - NaChar)) / Float64(NdChar * Float64(NdChar * NdChar)))));
	elseif (Vef <= 1.2e+268)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(0.5 * Float64(-1.0 / Float64(Float64(NdChar / Float64(NaChar * NaChar)) + Float64(-1.0 / NaChar))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Vef <= -1.55e+208)
		tmp = 0.5 * (-1.0 / ((NaChar * (NdChar - NaChar)) / (NdChar * (NdChar * NdChar))));
	elseif (Vef <= 1.2e+268)
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = 0.5 * (-1.0 / ((NdChar / (NaChar * NaChar)) + (-1.0 / NaChar)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Vef, -1.55e+208], N[(0.5 * N[(-1.0 / N[(N[(NaChar * N[(NdChar - NaChar), $MachinePrecision]), $MachinePrecision] / N[(NdChar * N[(NdChar * NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 1.2e+268], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-1.0 / N[(N[(NdChar / N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -1.5499999999999999e208

   1. Initial program 100.0%

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

   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 14.0%

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

   7. Simplified 14.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   8. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{NaChar}^{3} + {NdChar}^{3}}{\color{blue}{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{\color{blue}{\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \color{blue}{\left({NaChar}^{3} + {NdChar}^{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(NaChar \cdot NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left(\color{blue}{{NaChar}^{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left({\color{blue}{NaChar}}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \left({NaChar}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left({NaChar}^{3}\right), \color{blue}{\left({NdChar}^{3}\right)}\right)\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left(NaChar \cdot \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{*.f64}\left(NaChar, NaChar\right)\right), \left({NdChar}^{3}\right)\right)\right)\right)\right) \]

      14. cube-mult N/A

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

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

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

      16. *-lowering-*.f64 7.2%

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

   9. Applied egg-rr 7.2%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NaChar \cdot NaChar + NdChar \cdot \left(NdChar - NaChar\right)}{NaChar \cdot \left(NaChar \cdot NaChar\right) + NdChar \cdot \left(NdChar \cdot NdChar\right)}}} \]

   10. Taylor expanded in NdChar around 0

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. mul-1-neg N/A

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

       4. unpow2 N/A

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

       5. distribute-lft-in N/A

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

       6. +-commutative N/A

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

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

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

       8. mul-1-neg N/A

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

       9. unsub-neg N/A

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

       10. --lowering--.f64 7.5%

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

   12. Simplified 7.5%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{NaChar \cdot \left(NaChar - NdChar\right)}}{NaChar \cdot \left(NaChar \cdot NaChar\right) + NdChar \cdot \left(NdChar \cdot NdChar\right)}} \]

   13. Taylor expanded in NaChar around 0

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{\_.f64}\left(NaChar, NdChar\right)\right), \color{blue}{\left({NdChar}^{3}\right)}\right)\right)\right) \]
   14. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 31.8%

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

   15. Simplified 31.8%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{NaChar \cdot \left(NaChar - NdChar\right)}{\color{blue}{NdChar \cdot \left(NdChar \cdot NdChar\right)}}} \]

   if -1.5499999999999999e208 < Vef < 1.2e268

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 32.3%

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

   7. Simplified 32.3%

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

   if 1.2e268 < 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 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 8.3%

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

   7. Simplified 8.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   8. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{NaChar}^{3} + {NdChar}^{3}}{\color{blue}{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{\color{blue}{\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \color{blue}{\left({NaChar}^{3} + {NdChar}^{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(NaChar \cdot NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left(\color{blue}{{NaChar}^{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left({\color{blue}{NaChar}}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \left({NaChar}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left({NaChar}^{3}\right), \color{blue}{\left({NdChar}^{3}\right)}\right)\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left(NaChar \cdot \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{*.f64}\left(NaChar, NaChar\right)\right), \left({NdChar}^{3}\right)\right)\right)\right)\right) \]

      14. cube-mult N/A

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

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

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

      16. *-lowering-*.f64 5.3%

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

   9. Applied egg-rr 5.3%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NaChar \cdot NaChar + NdChar \cdot \left(NdChar - NaChar\right)}{NaChar \cdot \left(NaChar \cdot NaChar\right) + NdChar \cdot \left(NdChar \cdot NdChar\right)}}} \]

   10. Taylor expanded in NdChar around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 58.3%

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

   12. Simplified 58.3%

       \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{NaChar} - \frac{NdChar}{NaChar \cdot NaChar}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.55 \cdot 10^{+208}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{NaChar \cdot \left(NdChar - NaChar\right)}{NdChar \cdot \left(NdChar \cdot NdChar\right)}}\\ \mathbf{elif}\;Vef \leq 1.2 \cdot 10^{+268}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{NdChar}{NaChar \cdot NaChar} + \frac{-1}{NaChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq 2.4 \cdot 10^{+265}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{NdChar}{NaChar \cdot NaChar} + \frac{-1}{NaChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Vef 2.4e+265)
   (* 0.5 (+ NdChar NaChar))
   (* 0.5 (/ -1.0 (+ (/ NdChar (* NaChar NaChar)) (/ -1.0 NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Vef <= 2.4e+265) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * (-1.0 / ((NdChar / (NaChar * NaChar)) + (-1.0 / NaChar)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (vef <= 2.4d+265) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = 0.5d0 * ((-1.0d0) / ((ndchar / (nachar * nachar)) + ((-1.0d0) / nachar)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Vef <= 2.4e+265) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * (-1.0 / ((NdChar / (NaChar * NaChar)) + (-1.0 / NaChar)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Vef <= 2.4e+265:
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = 0.5 * (-1.0 / ((NdChar / (NaChar * NaChar)) + (-1.0 / NaChar)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Vef <= 2.4e+265)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(0.5 * Float64(-1.0 / Float64(Float64(NdChar / Float64(NaChar * NaChar)) + Float64(-1.0 / NaChar))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Vef <= 2.4e+265)
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = 0.5 * (-1.0 / ((NdChar / (NaChar * NaChar)) + (-1.0 / NaChar)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Vef, 2.4e+265], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-1.0 / N[(N[(NdChar / N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Vef < 2.4e265

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 30.6%

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

   7. Simplified 30.6%

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

   if 2.4e265 < 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 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 8.3%

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

   7. Simplified 8.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   8. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{NaChar}^{3} + {NdChar}^{3}}{\color{blue}{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{\color{blue}{\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \color{blue}{\left({NaChar}^{3} + {NdChar}^{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(NaChar \cdot NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left(\color{blue}{{NaChar}^{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left({\color{blue}{NaChar}}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \left({NaChar}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left({NaChar}^{3}\right), \color{blue}{\left({NdChar}^{3}\right)}\right)\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left(NaChar \cdot \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{*.f64}\left(NaChar, NaChar\right)\right), \left({NdChar}^{3}\right)\right)\right)\right)\right) \]

      14. cube-mult N/A

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

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

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

      16. *-lowering-*.f64 5.3%

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

   9. Applied egg-rr 5.3%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NaChar \cdot NaChar + NdChar \cdot \left(NdChar - NaChar\right)}{NaChar \cdot \left(NaChar \cdot NaChar\right) + NdChar \cdot \left(NdChar \cdot NdChar\right)}}} \]

   10. Taylor expanded in NdChar around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 58.3%

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

   12. Simplified 58.3%

       \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1}{NaChar} - \frac{NdChar}{NaChar \cdot NaChar}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq 2.4 \cdot 10^{+265}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{NdChar}{NaChar \cdot NaChar} + \frac{-1}{NaChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq 4.85 \cdot 10^{+260}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{-1 + \frac{NdChar}{NaChar}}{NaChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Vef 4.85e+260)
   (* 0.5 (+ NdChar NaChar))
   (* 0.5 (/ -1.0 (/ (+ -1.0 (/ NdChar NaChar)) NaChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Vef <= 4.85e+260) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * (-1.0 / ((-1.0 + (NdChar / NaChar)) / NaChar));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (vef <= 4.85d+260) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = 0.5d0 * ((-1.0d0) / (((-1.0d0) + (ndchar / nachar)) / nachar))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Vef <= 4.85e+260) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * (-1.0 / ((-1.0 + (NdChar / NaChar)) / NaChar));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Vef <= 4.85e+260:
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = 0.5 * (-1.0 / ((-1.0 + (NdChar / NaChar)) / NaChar))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Vef <= 4.85e+260)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(0.5 * Float64(-1.0 / Float64(Float64(-1.0 + Float64(NdChar / NaChar)) / NaChar)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Vef <= 4.85e+260)
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = 0.5 * (-1.0 / ((-1.0 + (NdChar / NaChar)) / NaChar));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Vef, 4.85e+260], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-1.0 / N[(N[(-1.0 + N[(NdChar / NaChar), $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Vef < 4.8500000000000002e260

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 30.8%

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

   7. Simplified 30.8%

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

   if 4.8500000000000002e260 < 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 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 8.8%

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

   7. Simplified 8.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   8. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{NaChar}^{3} + {NdChar}^{3}}{\color{blue}{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{\color{blue}{\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)}{{NaChar}^{3} + {NdChar}^{3}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(NaChar \cdot NaChar + \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \color{blue}{\left({NaChar}^{3} + {NdChar}^{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(NaChar \cdot NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left(\color{blue}{{NaChar}^{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot NdChar - NaChar \cdot NdChar\right)\right), \left({\color{blue}{NaChar}}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \left(NdChar \cdot \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \left(NdChar - NaChar\right)\right)\right), \left({NaChar}^{\color{blue}{3}} + {NdChar}^{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \left({NaChar}^{3} + {NdChar}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left({NaChar}^{3}\right), \color{blue}{\left({NdChar}^{3}\right)}\right)\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\left(NaChar \cdot \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \left(NaChar \cdot NaChar\right)\right), \left({\color{blue}{NdChar}}^{3}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, NaChar\right), \mathsf{*.f64}\left(NdChar, \mathsf{\_.f64}\left(NdChar, NaChar\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{*.f64}\left(NaChar, NaChar\right)\right), \left({NdChar}^{3}\right)\right)\right)\right)\right) \]

      14. cube-mult N/A

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

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

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

      16. *-lowering-*.f64 6.3%

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

   9. Applied egg-rr 6.3%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NaChar \cdot NaChar + NdChar \cdot \left(NdChar - NaChar\right)}{NaChar \cdot \left(NaChar \cdot NaChar\right) + NdChar \cdot \left(NdChar \cdot NdChar\right)}}} \]

   10. Taylor expanded in NaChar around inf

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

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

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

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

       5. /-lowering-/.f64 49.7%

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

   12. Simplified 49.7%

       \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{1 - \frac{NdChar}{NaChar}}{NaChar}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq 4.85 \cdot 10^{+260}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{-1}{\frac{-1 + \frac{NdChar}{NaChar}}{NaChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 28.0% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq 3.8 \cdot 10^{+270}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(NaChar \cdot NaChar\right) \cdot \frac{-1}{NdChar - NaChar}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Vef 3.8e+270)
   (* 0.5 (+ NdChar NaChar))
   (* 0.5 (* (* NaChar NaChar) (/ -1.0 (- NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Vef <= 3.8e+270) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * ((NaChar * NaChar) * (-1.0 / (NdChar - NaChar)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (vef <= 3.8d+270) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = 0.5d0 * ((nachar * nachar) * ((-1.0d0) / (ndchar - nachar)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Vef <= 3.8e+270) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * ((NaChar * NaChar) * (-1.0 / (NdChar - NaChar)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Vef <= 3.8e+270:
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = 0.5 * ((NaChar * NaChar) * (-1.0 / (NdChar - NaChar)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Vef <= 3.8e+270)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(0.5 * Float64(Float64(NaChar * NaChar) * Float64(-1.0 / Float64(NdChar - NaChar))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Vef <= 3.8e+270)
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = 0.5 * ((NaChar * NaChar) * (-1.0 / (NdChar - NaChar)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Vef, 3.8e+270], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(NaChar * NaChar), $MachinePrecision] * N[(-1.0 / N[(NdChar - NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Vef < 3.80000000000000018e270

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 30.6%

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

   7. Simplified 30.6%

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

   if 3.80000000000000018e270 < 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 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 8.3%

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

   7. Simplified 8.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      8. --lowering--.f64 18.0%

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

   9. Applied egg-rr 18.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{NaChar - NdChar}\right)} \]

   10. Taylor expanded in NaChar around inf

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\color{blue}{\left({NaChar}^{2}\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(NaChar, NdChar\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

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

       2. *-lowering-*.f64 58.1%

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

   12. Simplified 58.1%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq 3.8 \cdot 10^{+270}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(NaChar \cdot NaChar\right) \cdot \frac{-1}{NdChar - NaChar}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 21.7% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -1.26 \cdot 10^{+138}:\\ \;\;\;\;\frac{NaChar}{2}\\ \mathbf{elif}\;NaChar \leq 2.15 \cdot 10^{+181}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -1.26e+138)
   (/ NaChar 2.0)
   (if (<= NaChar 2.15e+181) (* NdChar 0.5) (/ NaChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.26e+138) {
		tmp = NaChar / 2.0;
	} else if (NaChar <= 2.15e+181) {
		tmp = NdChar * 0.5;
	} else {
		tmp = NaChar / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-1.26d+138)) then
        tmp = nachar / 2.0d0
    else if (nachar <= 2.15d+181) then
        tmp = ndchar * 0.5d0
    else
        tmp = nachar / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.26e+138) {
		tmp = NaChar / 2.0;
	} else if (NaChar <= 2.15e+181) {
		tmp = NdChar * 0.5;
	} else {
		tmp = NaChar / 2.0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -1.26e+138:
		tmp = NaChar / 2.0
	elif NaChar <= 2.15e+181:
		tmp = NdChar * 0.5
	else:
		tmp = NaChar / 2.0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -1.26e+138)
		tmp = Float64(NaChar / 2.0);
	elseif (NaChar <= 2.15e+181)
		tmp = Float64(NdChar * 0.5);
	else
		tmp = Float64(NaChar / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -1.26e+138)
		tmp = NaChar / 2.0;
	elseif (NaChar <= 2.15e+181)
		tmp = NdChar * 0.5;
	else
		tmp = NaChar / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -1.26e+138], N[(NaChar / 2.0), $MachinePrecision], If[LessEqual[NaChar, 2.15e+181], N[(NdChar * 0.5), $MachinePrecision], N[(NaChar / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.25999999999999994e138 or 2.14999999999999986e181 < 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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{2}\right) \]
   9. Step-by-step derivation

   10. Simplified 29.0%

       \[\leadsto \frac{NaChar}{\color{blue}{2}} \]

   if -1.25999999999999994e138 < NaChar < 2.14999999999999986e181

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 29.7%

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

   7. Simplified 29.7%

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

   8. Taylor expanded in NaChar around 0

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

      1. *-lowering-*.f64 25.6%

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

   10. Simplified 25.6%

       \[\leadsto \color{blue}{0.5 \cdot NdChar} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.26 \cdot 10^{+138}:\\ \;\;\;\;\frac{NaChar}{2}\\ \mathbf{elif}\;NaChar \leq 2.15 \cdot 10^{+181}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.9% 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 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
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 29.8%

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

7. Simplified 29.8%

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

8. Final simplification 29.8%

   \[\leadsto 0.5 \cdot \left(NdChar + NaChar\right) \]
9. Add Preprocessing

Alternative 20: 18.6% accurate, 76.3× speedup

Math

\[\begin{array}{l} \\ NdChar \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
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 29.8%

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

7. Simplified 29.8%

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

8. Taylor expanded in NaChar around 0

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

   1. *-lowering-*.f64 21.1%

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

10. Simplified 21.1%

    \[\leadsto \color{blue}{0.5 \cdot NdChar} \]

11. Final simplification 21.1%

    \[\leadsto NdChar \cdot 0.5 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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))))))