Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 12.5s

Alternatives: 24

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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) - 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]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 67.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_1 := \frac{NaChar}{1 + t\_0}\\ t_2 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_3 := t\_2 + t\_1\\ t_4 := t\_2 + 0.5 \cdot NaChar\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot NdChar + t\_1\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{NaChar}{t\_0 + 1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_1 (/ NaChar (+ 1.0 t_0)))
        (t_2 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_3 (+ t_2 t_1))
        (t_4 (+ t_2 (* 0.5 NaChar))))
   (if (<= t_3 -4e+133)
     (+ (* 0.5 NdChar) t_1)
     (if (<= t_3 -2e-180)
       t_4
       (if (<= t_3 5e-88)
         (/ NaChar (+ t_0 1.0))
         (if (<= t_3 5e+98)
           (+ (/ NdChar (+ 2.0 (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))) t_1)
           t_4))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = NaChar / (1.0 + t_0);
	double t_2 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_3 = t_2 + t_1;
	double t_4 = t_2 + (0.5 * NaChar);
	double tmp;
	if (t_3 <= -4e+133) {
		tmp = (0.5 * NdChar) + t_1;
	} else if (t_3 <= -2e-180) {
		tmp = t_4;
	} else if (t_3 <= 5e-88) {
		tmp = NaChar / (t_0 + 1.0);
	} else if (t_3 <= 5e+98) {
		tmp = (NdChar / (2.0 + ((((mu + Vef) + EDonor) - Ec) / KbT))) + t_1;
	} else {
		tmp = t_4;
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_1 = nachar / (1.0d0 + t_0)
    t_2 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_3 = t_2 + t_1
    t_4 = t_2 + (0.5d0 * nachar)
    if (t_3 <= (-4d+133)) then
        tmp = (0.5d0 * ndchar) + t_1
    else if (t_3 <= (-2d-180)) then
        tmp = t_4
    else if (t_3 <= 5d-88) then
        tmp = nachar / (t_0 + 1.0d0)
    else if (t_3 <= 5d+98) then
        tmp = (ndchar / (2.0d0 + ((((mu + vef) + edonor) - ec) / kbt))) + t_1
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = NaChar / (1.0 + t_0);
	double t_2 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_3 = t_2 + t_1;
	double t_4 = t_2 + (0.5 * NaChar);
	double tmp;
	if (t_3 <= -4e+133) {
		tmp = (0.5 * NdChar) + t_1;
	} else if (t_3 <= -2e-180) {
		tmp = t_4;
	} else if (t_3 <= 5e-88) {
		tmp = NaChar / (t_0 + 1.0);
	} else if (t_3 <= 5e+98) {
		tmp = (NdChar / (2.0 + ((((mu + Vef) + EDonor) - Ec) / KbT))) + t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = NaChar / (1.0 + t_0)
	t_2 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_3 = t_2 + t_1
	t_4 = t_2 + (0.5 * NaChar)
	tmp = 0
	if t_3 <= -4e+133:
		tmp = (0.5 * NdChar) + t_1
	elif t_3 <= -2e-180:
		tmp = t_4
	elif t_3 <= 5e-88:
		tmp = NaChar / (t_0 + 1.0)
	elif t_3 <= 5e+98:
		tmp = (NdChar / (2.0 + ((((mu + Vef) + EDonor) - Ec) / KbT))) + t_1
	else:
		tmp = t_4
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = Float64(NaChar / Float64(1.0 + t_0))
	t_2 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_3 = Float64(t_2 + t_1)
	t_4 = Float64(t_2 + Float64(0.5 * NaChar))
	tmp = 0.0
	if (t_3 <= -4e+133)
		tmp = Float64(Float64(0.5 * NdChar) + t_1);
	elseif (t_3 <= -2e-180)
		tmp = t_4;
	elseif (t_3 <= 5e-88)
		tmp = Float64(NaChar / Float64(t_0 + 1.0));
	elseif (t_3 <= 5e+98)
		tmp = Float64(Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT))) + t_1);
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_1 = NaChar / (1.0 + t_0);
	t_2 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_3 = t_2 + t_1;
	t_4 = t_2 + (0.5 * NaChar);
	tmp = 0.0;
	if (t_3 <= -4e+133)
		tmp = (0.5 * NdChar) + t_1;
	elseif (t_3 <= -2e-180)
		tmp = t_4;
	elseif (t_3 <= 5e-88)
		tmp = NaChar / (t_0 + 1.0);
	elseif (t_3 <= 5e+98)
		tmp = (NdChar / (2.0 + ((((mu + Vef) + EDonor) - Ec) / KbT))) + t_1;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+133], N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -2e-180], t$95$4, If[LessEqual[t$95$3, 5e-88], N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+98], N[(N[(NdChar / N[(2.0 + N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$4]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.0000000000000001e133

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -4.0000000000000001e133 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-180 or 4.9999999999999998e98 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if -2e-180 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5.00000000000000009e-88

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if 5.00000000000000009e-88 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999998e98

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-add N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 82.6

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

   5. Applied rewrites 82.6%

      \[\leadsto \frac{NdChar}{\color{blue}{2 + \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -4 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + 0.5 \cdot NaChar\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + 0.5 \cdot NaChar\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_2 := \frac{NaChar}{1 + t\_1}\\ t_3 := t\_0 + t\_2\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-232}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{-Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{NaChar}{t\_1 + 1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 + 0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_2 (/ NaChar (+ 1.0 t_1)))
        (t_3 (+ t_0 t_2)))
   (if (<= t_3 -1e-232)
     (+
      (/ NdChar (+ (exp (/ (- Ec) KbT)) 1.0))
      (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))
     (if (<= t_3 5e-88)
       (/ NaChar (+ t_1 1.0))
       (if (<= t_3 5e+98)
         (+ (/ NdChar (+ 2.0 (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))) t_2)
         (+ t_0 (* 0.5 NaChar)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = NaChar / (1.0 + t_1);
	double t_3 = t_0 + t_2;
	double tmp;
	if (t_3 <= -1e-232) {
		tmp = (NdChar / (exp((-Ec / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (t_3 <= 5e-88) {
		tmp = NaChar / (t_1 + 1.0);
	} else if (t_3 <= 5e+98) {
		tmp = (NdChar / (2.0 + ((((mu + Vef) + EDonor) - Ec) / KbT))) + t_2;
	} else {
		tmp = t_0 + (0.5 * 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_1 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_2 = nachar / (1.0d0 + t_1)
    t_3 = t_0 + t_2
    if (t_3 <= (-1d-232)) then
        tmp = (ndchar / (exp((-ec / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    else if (t_3 <= 5d-88) then
        tmp = nachar / (t_1 + 1.0d0)
    else if (t_3 <= 5d+98) then
        tmp = (ndchar / (2.0d0 + ((((mu + vef) + edonor) - ec) / kbt))) + t_2
    else
        tmp = t_0 + (0.5d0 * 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 t_0 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = NaChar / (1.0 + t_1);
	double t_3 = t_0 + t_2;
	double tmp;
	if (t_3 <= -1e-232) {
		tmp = (NdChar / (Math.exp((-Ec / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (t_3 <= 5e-88) {
		tmp = NaChar / (t_1 + 1.0);
	} else if (t_3 <= 5e+98) {
		tmp = (NdChar / (2.0 + ((((mu + Vef) + EDonor) - Ec) / KbT))) + t_2;
	} else {
		tmp = t_0 + (0.5 * NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = NaChar / (1.0 + t_1)
	t_3 = t_0 + t_2
	tmp = 0
	if t_3 <= -1e-232:
		tmp = (NdChar / (math.exp((-Ec / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	elif t_3 <= 5e-88:
		tmp = NaChar / (t_1 + 1.0)
	elif t_3 <= 5e+98:
		tmp = (NdChar / (2.0 + ((((mu + Vef) + EDonor) - Ec) / KbT))) + t_2
	else:
		tmp = t_0 + (0.5 * NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_1 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = Float64(NaChar / Float64(1.0 + t_1))
	t_3 = Float64(t_0 + t_2)
	tmp = 0.0
	if (t_3 <= -1e-232)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(-Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	elseif (t_3 <= 5e-88)
		tmp = Float64(NaChar / Float64(t_1 + 1.0));
	elseif (t_3 <= 5e+98)
		tmp = Float64(Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT))) + t_2);
	else
		tmp = Float64(t_0 + Float64(0.5 * NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_2 = NaChar / (1.0 + t_1);
	t_3 = t_0 + t_2;
	tmp = 0.0;
	if (t_3 <= -1e-232)
		tmp = (NdChar / (exp((-Ec / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	elseif (t_3 <= 5e-88)
		tmp = NaChar / (t_1 + 1.0);
	elseif (t_3 <= 5e+98)
		tmp = (NdChar / (2.0 + ((((mu + Vef) + EDonor) - Ec) / KbT))) + t_2;
	else
		tmp = t_0 + (0.5 * NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-232], N[(N[(NdChar / N[(N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e-88], N[(NaChar / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+98], N[(N[(NdChar / N[(2.0 + N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(t$95$0 + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.00000000000000002e-232

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in Vef around 0

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in EDonor around 0

      \[\leadsto \frac{NdChar}{1 + e^{-1 \cdot \frac{Ec}{KbT}}} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.4%

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

   if -1.00000000000000002e-232 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5.00000000000000009e-88

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if 5.00000000000000009e-88 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999998e98

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-add N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if 4.9999999999999998e98 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-232}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{-Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + 0.5 \cdot NaChar\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_2 := \frac{NaChar}{1 + t\_1}\\ t_3 := t\_0 + t\_2\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot NdChar + t\_2\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-180} \lor \neg \left(t\_3 \leq 5 \cdot 10^{+79}\right):\\ \;\;\;\;t\_0 + 0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_1 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_2 (/ NaChar (+ 1.0 t_1)))
        (t_3 (+ t_0 t_2)))
   (if (<= t_3 -4e+133)
     (+ (* 0.5 NdChar) t_2)
     (if (or (<= t_3 -2e-180) (not (<= t_3 5e+79)))
       (+ t_0 (* 0.5 NaChar))
       (/ NaChar (+ t_1 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = NaChar / (1.0 + t_1);
	double t_3 = t_0 + t_2;
	double tmp;
	if (t_3 <= -4e+133) {
		tmp = (0.5 * NdChar) + t_2;
	} else if ((t_3 <= -2e-180) || !(t_3 <= 5e+79)) {
		tmp = t_0 + (0.5 * NaChar);
	} else {
		tmp = NaChar / (t_1 + 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_1 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_2 = nachar / (1.0d0 + t_1)
    t_3 = t_0 + t_2
    if (t_3 <= (-4d+133)) then
        tmp = (0.5d0 * ndchar) + t_2
    else if ((t_3 <= (-2d-180)) .or. (.not. (t_3 <= 5d+79))) then
        tmp = t_0 + (0.5d0 * nachar)
    else
        tmp = nachar / (t_1 + 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 t_0 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = NaChar / (1.0 + t_1);
	double t_3 = t_0 + t_2;
	double tmp;
	if (t_3 <= -4e+133) {
		tmp = (0.5 * NdChar) + t_2;
	} else if ((t_3 <= -2e-180) || !(t_3 <= 5e+79)) {
		tmp = t_0 + (0.5 * NaChar);
	} else {
		tmp = NaChar / (t_1 + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = NaChar / (1.0 + t_1)
	t_3 = t_0 + t_2
	tmp = 0
	if t_3 <= -4e+133:
		tmp = (0.5 * NdChar) + t_2
	elif (t_3 <= -2e-180) or not (t_3 <= 5e+79):
		tmp = t_0 + (0.5 * NaChar)
	else:
		tmp = NaChar / (t_1 + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_1 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = Float64(NaChar / Float64(1.0 + t_1))
	t_3 = Float64(t_0 + t_2)
	tmp = 0.0
	if (t_3 <= -4e+133)
		tmp = Float64(Float64(0.5 * NdChar) + t_2);
	elseif ((t_3 <= -2e-180) || !(t_3 <= 5e+79))
		tmp = Float64(t_0 + Float64(0.5 * NaChar));
	else
		tmp = Float64(NaChar / Float64(t_1 + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_2 = NaChar / (1.0 + t_1);
	t_3 = t_0 + t_2;
	tmp = 0.0;
	if (t_3 <= -4e+133)
		tmp = (0.5 * NdChar) + t_2;
	elseif ((t_3 <= -2e-180) || ~((t_3 <= 5e+79)))
		tmp = t_0 + (0.5 * NaChar);
	else
		tmp = NaChar / (t_1 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+133], N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$2), $MachinePrecision], If[Or[LessEqual[t$95$3, -2e-180], N[Not[LessEqual[t$95$3, 5e+79]], $MachinePrecision]], N[(t$95$0 + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.0000000000000001e133

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -4.0000000000000001e133 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-180 or 5e79 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if -2e-180 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5e79

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 76.3

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

   5. Applied rewrites 76.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -4 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-180} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + 0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_1 := \frac{NaChar}{1 + t\_0}\\ t_2 := 0.5 \cdot NdChar + t\_1\\ t_3 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + t\_1\\ \mathbf{if}\;t\_3 \leq -10:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + 0.5 \cdot NaChar\\ \mathbf{elif}\;t\_3 \leq 10^{-94}:\\ \;\;\;\;\frac{NaChar}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_1 (/ NaChar (+ 1.0 t_0)))
        (t_2 (+ (* 0.5 NdChar) t_1))
        (t_3
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          t_1)))
   (if (<= t_3 -10.0)
     t_2
     (if (<= t_3 -2e-180)
       (+ (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))) (* 0.5 NaChar))
       (if (<= t_3 1e-94) (/ NaChar (+ t_0 1.0)) 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 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = NaChar / (1.0 + t_0);
	double t_2 = (0.5 * NdChar) + t_1;
	double t_3 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_1;
	double tmp;
	if (t_3 <= -10.0) {
		tmp = t_2;
	} else if (t_3 <= -2e-180) {
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (0.5 * NaChar);
	} else if (t_3 <= 1e-94) {
		tmp = NaChar / (t_0 + 1.0);
	} 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) :: t_3
    real(8) :: tmp
    t_0 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_1 = nachar / (1.0d0 + t_0)
    t_2 = (0.5d0 * ndchar) + t_1
    t_3 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + t_1
    if (t_3 <= (-10.0d0)) then
        tmp = t_2
    else if (t_3 <= (-2d-180)) then
        tmp = (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt)))) + (0.5d0 * nachar)
    else if (t_3 <= 1d-94) then
        tmp = nachar / (t_0 + 1.0d0)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = NaChar / (1.0 + t_0);
	double t_2 = (0.5 * NdChar) + t_1;
	double t_3 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_1;
	double tmp;
	if (t_3 <= -10.0) {
		tmp = t_2;
	} else if (t_3 <= -2e-180) {
		tmp = (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT)))) + (0.5 * NaChar);
	} else if (t_3 <= 1e-94) {
		tmp = NaChar / (t_0 + 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = NaChar / (1.0 + t_0)
	t_2 = (0.5 * NdChar) + t_1
	t_3 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_1
	tmp = 0
	if t_3 <= -10.0:
		tmp = t_2
	elif t_3 <= -2e-180:
		tmp = (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT)))) + (0.5 * NaChar)
	elif t_3 <= 1e-94:
		tmp = NaChar / (t_0 + 1.0)
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = Float64(NaChar / Float64(1.0 + t_0))
	t_2 = Float64(Float64(0.5 * NdChar) + t_1)
	t_3 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + t_1)
	tmp = 0.0
	if (t_3 <= -10.0)
		tmp = t_2;
	elseif (t_3 <= -2e-180)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))) + Float64(0.5 * NaChar));
	elseif (t_3 <= 1e-94)
		tmp = Float64(NaChar / Float64(t_0 + 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_1 = NaChar / (1.0 + t_0);
	t_2 = (0.5 * NdChar) + t_1;
	t_3 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_1;
	tmp = 0.0;
	if (t_3 <= -10.0)
		tmp = t_2;
	elseif (t_3 <= -2e-180)
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (0.5 * NaChar);
	elseif (t_3 <= 1e-94)
		tmp = NaChar / (t_0 + 1.0);
	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[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -10.0], t$95$2, If[LessEqual[t$95$3, -2e-180], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-94], N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -10 or 9.9999999999999996e-95 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   if -10 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-180

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in EDonor around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 68.3

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

   8. Applied rewrites 68.3%

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

   if -2e-180 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999996e-95

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 82.0

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

   5. Applied rewrites 82.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -10:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + 0.5 \cdot NaChar\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 10^{-94}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-304} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-228}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 t_0)))))
   (if (or (<= t_1 -1e-304) (not (<= t_1 2e-228)))
     (+
      (/ NdChar (+ (exp (/ (- EDonor Ec) KbT)) 1.0))
      (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))
     (/ NaChar (+ t_0 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	double tmp;
	if ((t_1 <= -1e-304) || !(t_1 <= 2e-228)) {
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	} else {
		tmp = NaChar / (t_0 + 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + t_0))
    if ((t_1 <= (-1d-304)) .or. (.not. (t_1 <= 2d-228))) then
        tmp = (ndchar / (exp(((edonor - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    else
        tmp = nachar / (t_0 + 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 t_0 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	double tmp;
	if ((t_1 <= -1e-304) || !(t_1 <= 2e-228)) {
		tmp = (NdChar / (Math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	} else {
		tmp = NaChar / (t_0 + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0))
	tmp = 0
	if (t_1 <= -1e-304) or not (t_1 <= 2e-228):
		tmp = (NdChar / (math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	else:
		tmp = NaChar / (t_0 + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + t_0)))
	tmp = 0.0
	if ((t_1 <= -1e-304) || !(t_1 <= 2e-228))
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	else
		tmp = Float64(NaChar / Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	tmp = 0.0;
	if ((t_1 <= -1e-304) || ~((t_1 <= 2e-228)))
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	else
		tmp = NaChar / (t_0 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-304], N[Not[LessEqual[t$95$1, 2e-228]], $MachinePrecision]], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999971e-305 or 2.00000000000000007e-228 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in Vef around 0

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

   8. Applied rewrites 76.4%

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

   if -9.99999999999999971e-305 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.00000000000000007e-228

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-304} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-228}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-304} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-267}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{EAccept + \left(Ev + Vef\right)}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -1e-304) (not (<= t_0 5e-267)))
     (* 0.5 (+ NaChar NdChar))
     (/ NaChar (+ 2.0 (/ (+ EAccept (+ Ev Vef)) KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -1e-304) || !(t_0 <= 5e-267)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (2.0 + ((EAccept + (Ev + Vef)) / KbT));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-1d-304)) .or. (.not. (t_0 <= 5d-267))) then
        tmp = 0.5d0 * (nachar + ndchar)
    else
        tmp = nachar / (2.0d0 + ((eaccept + (ev + vef)) / kbt))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -1e-304) || !(t_0 <= 5e-267)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (2.0 + ((EAccept + (Ev + Vef)) / KbT));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -1e-304) or not (t_0 <= 5e-267):
		tmp = 0.5 * (NaChar + NdChar)
	else:
		tmp = NaChar / (2.0 + ((EAccept + (Ev + Vef)) / KbT))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 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) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -1e-304) || !(t_0 <= 5e-267))
		tmp = Float64(0.5 * Float64(NaChar + NdChar));
	else
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(EAccept + Float64(Ev + Vef)) / KbT)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -1e-304) || ~((t_0 <= 5e-267)))
		tmp = 0.5 * (NaChar + NdChar);
	else
		tmp = NaChar / (2.0 + ((EAccept + (Ev + Vef)) / KbT));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-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]}, If[Or[LessEqual[t$95$0, -1e-304], N[Not[LessEqual[t$95$0, 5e-267]], $MachinePrecision]], N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(2.0 + N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999971e-305 or 4.9999999999999999e-267 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 36.9

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

   5. Applied rewrites 36.9%

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

   if -9.99999999999999971e-305 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999999e-267

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in KbT around inf

      \[\leadsto \frac{NaChar}{2 + \left(\frac{EAccept}{KbT} + \color{blue}{\left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.3%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-304} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-267}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{EAccept + \left(Ev + Vef\right)}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{if}\;Vef \leq -1.06 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq -6.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 3.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + t\_0\\ \mathbf{elif}\;Vef \leq 3.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))
        (t_1 (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0)))
   (if (<= Vef -1.06e+127)
     t_1
     (if (<= Vef -6.2e-186)
       (+
        (/ NdChar (+ (exp (/ (- EDonor Ec) KbT)) 1.0))
        (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))
       (if (<= Vef 3.1e-29)
         (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_0)
         (if (<= Vef 3.5e+114)
           (+
            (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
            (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
           t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	double tmp;
	if (Vef <= -1.06e+127) {
		tmp = t_1;
	} else if (Vef <= -6.2e-186) {
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (Vef <= 3.1e-29) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	} else if (Vef <= 3.5e+114) {
		tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((vef / kbt)))) + t_0
    if (vef <= (-1.06d+127)) then
        tmp = t_1
    else if (vef <= (-6.2d-186)) then
        tmp = (ndchar / (exp(((edonor - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    else if (vef <= 3.1d-29) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + t_0
    else if (vef <= 3.5d+114) then
        tmp = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp((ev / kbt))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	double tmp;
	if (Vef <= -1.06e+127) {
		tmp = t_1;
	} else if (Vef <= -6.2e-186) {
		tmp = (NdChar / (Math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (Vef <= 3.1e-29) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + t_0;
	} else if (Vef <= 3.5e+114) {
		tmp = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	tmp = 0
	if Vef <= -1.06e+127:
		tmp = t_1
	elif Vef <= -6.2e-186:
		tmp = (NdChar / (math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	elif Vef <= 3.1e-29:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + t_0
	elif Vef <= 3.5e+114:
		tmp = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0)
	tmp = 0.0
	if (Vef <= -1.06e+127)
		tmp = t_1;
	elseif (Vef <= -6.2e-186)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	elseif (Vef <= 3.1e-29)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + t_0);
	elseif (Vef <= 3.5e+114)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	tmp = 0.0;
	if (Vef <= -1.06e+127)
		tmp = t_1;
	elseif (Vef <= -6.2e-186)
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	elseif (Vef <= 3.1e-29)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	elseif (Vef <= 3.5e+114)
		tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[Vef, -1.06e+127], t$95$1, If[LessEqual[Vef, -6.2e-186], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 3.1e-29], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[Vef, 3.5e+114], 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[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -1.06000000000000006e127 or 3.5000000000000001e114 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around inf

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

      1. lower-/.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if -1.06000000000000006e127 < Vef < -6.20000000000000018e-186

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in Vef around 0

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

   8. Applied rewrites 84.1%

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

   if -6.20000000000000018e-186 < Vef < 3.10000000000000026e-29

   1. Initial program 100.0%

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

   3. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 86.5

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

   5. Applied rewrites 86.5%

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

   if 3.10000000000000026e-29 < Vef < 3.5000000000000001e114

   1. Initial program 99.9%

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

   3. Taylor expanded in Ev around inf

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

      1. lower-/.f64 95.6

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

   5. Applied rewrites 95.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.06 \cdot 10^{+127}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq -6.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 3.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 3.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{if}\;Vef \leq -1.06 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq -6.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 1.4 \cdot 10^{-163}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + t\_0\\ \mathbf{elif}\;Vef \leq 1.08 \cdot 10^{+114}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))
        (t_1 (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0)))
   (if (<= Vef -1.06e+127)
     t_1
     (if (<= Vef -6.2e-186)
       (+
        (/ NdChar (+ (exp (/ (- EDonor Ec) KbT)) 1.0))
        (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))
       (if (<= Vef 1.4e-163)
         (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_0)
         (if (<= Vef 1.08e+114)
           (+
            (/ NdChar (+ (exp (/ (- (+ mu EDonor) Ec) KbT)) 1.0))
            (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))
           t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	double tmp;
	if (Vef <= -1.06e+127) {
		tmp = t_1;
	} else if (Vef <= -6.2e-186) {
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (Vef <= 1.4e-163) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	} else if (Vef <= 1.08e+114) {
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((vef / kbt)))) + t_0
    if (vef <= (-1.06d+127)) then
        tmp = t_1
    else if (vef <= (-6.2d-186)) then
        tmp = (ndchar / (exp(((edonor - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    else if (vef <= 1.4d-163) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + t_0
    else if (vef <= 1.08d+114) then
        tmp = (ndchar / (exp((((mu + edonor) - ec) / kbt)) + 1.0d0)) + (nachar / (exp((eaccept / kbt)) + 1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	double tmp;
	if (Vef <= -1.06e+127) {
		tmp = t_1;
	} else if (Vef <= -6.2e-186) {
		tmp = (NdChar / (Math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (Vef <= 1.4e-163) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + t_0;
	} else if (Vef <= 1.08e+114) {
		tmp = (NdChar / (Math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	tmp = 0
	if Vef <= -1.06e+127:
		tmp = t_1
	elif Vef <= -6.2e-186:
		tmp = (NdChar / (math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	elif Vef <= 1.4e-163:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + t_0
	elif Vef <= 1.08e+114:
		tmp = (NdChar / (math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp((EAccept / KbT)) + 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0)
	tmp = 0.0
	if (Vef <= -1.06e+127)
		tmp = t_1;
	elseif (Vef <= -6.2e-186)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	elseif (Vef <= 1.4e-163)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + t_0);
	elseif (Vef <= 1.08e+114)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + EDonor) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	tmp = 0.0;
	if (Vef <= -1.06e+127)
		tmp = t_1;
	elseif (Vef <= -6.2e-186)
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	elseif (Vef <= 1.4e-163)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	elseif (Vef <= 1.08e+114)
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((EAccept / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[Vef, -1.06e+127], t$95$1, If[LessEqual[Vef, -6.2e-186], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 1.4e-163], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[Vef, 1.08e+114], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -1.06000000000000006e127 or 1.08000000000000004e114 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around inf

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

      1. lower-/.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if -1.06000000000000006e127 < Vef < -6.20000000000000018e-186

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in Vef around 0

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

   8. Applied rewrites 84.1%

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

   if -6.20000000000000018e-186 < Vef < 1.4e-163

   1. Initial program 100.0%

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

   3. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if 1.4e-163 < Vef < 1.08000000000000004e114

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 80.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.06 \cdot 10^{+127}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq -6.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 1.4 \cdot 10^{-163}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.08 \cdot 10^{+114}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Ev + Vef\right) + EAccept\\ t_1 := \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{t\_0}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -4.5 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq -6.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 1.4 \cdot 10^{-163}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{t\_0 - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.08 \cdot 10^{+114}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Ev Vef) EAccept))
        (t_1
         (+
          (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
          (/ NaChar (+ (exp (/ t_0 KbT)) 1.0)))))
   (if (<= Vef -4.5e+127)
     t_1
     (if (<= Vef -6.2e-186)
       (+
        (/ NdChar (+ (exp (/ (- EDonor Ec) KbT)) 1.0))
        (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))
       (if (<= Vef 1.4e-163)
         (+
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- t_0 mu) KbT)))))
         (if (<= Vef 1.08e+114)
           (+
            (/ NdChar (+ (exp (/ (- (+ mu EDonor) Ec) KbT)) 1.0))
            (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))
           t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Ev + Vef) + EAccept;
	double t_1 = (NdChar / (exp((Vef / KbT)) + 1.0)) + (NaChar / (exp((t_0 / KbT)) + 1.0));
	double tmp;
	if (Vef <= -4.5e+127) {
		tmp = t_1;
	} else if (Vef <= -6.2e-186) {
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (Vef <= 1.4e-163) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp(((t_0 - mu) / KbT))));
	} else if (Vef <= 1.08e+114) {
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (ev + vef) + eaccept
    t_1 = (ndchar / (exp((vef / kbt)) + 1.0d0)) + (nachar / (exp((t_0 / kbt)) + 1.0d0))
    if (vef <= (-4.5d+127)) then
        tmp = t_1
    else if (vef <= (-6.2d-186)) then
        tmp = (ndchar / (exp(((edonor - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    else if (vef <= 1.4d-163) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / (1.0d0 + exp(((t_0 - mu) / kbt))))
    else if (vef <= 1.08d+114) then
        tmp = (ndchar / (exp((((mu + edonor) - ec) / kbt)) + 1.0d0)) + (nachar / (exp((eaccept / kbt)) + 1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Ev + Vef) + EAccept;
	double t_1 = (NdChar / (Math.exp((Vef / KbT)) + 1.0)) + (NaChar / (Math.exp((t_0 / KbT)) + 1.0));
	double tmp;
	if (Vef <= -4.5e+127) {
		tmp = t_1;
	} else if (Vef <= -6.2e-186) {
		tmp = (NdChar / (Math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (Vef <= 1.4e-163) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / (1.0 + Math.exp(((t_0 - mu) / KbT))));
	} else if (Vef <= 1.08e+114) {
		tmp = (NdChar / (Math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (Ev + Vef) + EAccept
	t_1 = (NdChar / (math.exp((Vef / KbT)) + 1.0)) + (NaChar / (math.exp((t_0 / KbT)) + 1.0))
	tmp = 0
	if Vef <= -4.5e+127:
		tmp = t_1
	elif Vef <= -6.2e-186:
		tmp = (NdChar / (math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	elif Vef <= 1.4e-163:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / (1.0 + math.exp(((t_0 - mu) / KbT))))
	elif Vef <= 1.08e+114:
		tmp = (NdChar / (math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp((EAccept / KbT)) + 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Ev + Vef) + EAccept)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(t_0 / KbT)) + 1.0)))
	tmp = 0.0
	if (Vef <= -4.5e+127)
		tmp = t_1;
	elseif (Vef <= -6.2e-186)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	elseif (Vef <= 1.4e-163)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(t_0 - mu) / KbT)))));
	elseif (Vef <= 1.08e+114)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + EDonor) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (Ev + Vef) + EAccept;
	t_1 = (NdChar / (exp((Vef / KbT)) + 1.0)) + (NaChar / (exp((t_0 / KbT)) + 1.0));
	tmp = 0.0;
	if (Vef <= -4.5e+127)
		tmp = t_1;
	elseif (Vef <= -6.2e-186)
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	elseif (Vef <= 1.4e-163)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp(((t_0 - mu) / KbT))));
	elseif (Vef <= 1.08e+114)
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((EAccept / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(t$95$0 / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -4.5e+127], t$95$1, If[LessEqual[Vef, -6.2e-186], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 1.4e-163], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(t$95$0 - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 1.08e+114], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -4.50000000000000034e127 or 1.08000000000000004e114 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 82.5%

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

   if -4.50000000000000034e127 < Vef < -6.20000000000000018e-186

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in Vef around 0

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

   8. Applied rewrites 84.1%

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

   if -6.20000000000000018e-186 < Vef < 1.4e-163

   1. Initial program 100.0%

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

   3. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if 1.4e-163 < Vef < 1.08000000000000004e114

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 80.9%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -4.5 \cdot 10^{+127}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq -6.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 1.4 \cdot 10^{-163}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.08 \cdot 10^{+114}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;mu \leq -9.2 \cdot 10^{+29} \lor \neg \left(mu \leq 1.9 \cdot 10^{+192}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= mu -9.2e+29) (not (<= mu 1.9e+192)))
   (+
    (/ NdChar (+ (exp (/ (- (+ mu EDonor) Ec) KbT)) 1.0))
    (/ NaChar (+ (exp (/ (- (+ EAccept Ev) mu) KbT)) 1.0)))
   (+
    (/ NdChar (+ (exp (/ (- (+ EDonor Vef) Ec) KbT)) 1.0))
    (/ NaChar (+ (exp (/ (+ (+ Ev Vef) EAccept) 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 ((mu <= -9.2e+29) || !(mu <= 1.9e+192)) {
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	} else {
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / 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 ((mu <= (-9.2d+29)) .or. (.not. (mu <= 1.9d+192))) then
        tmp = (ndchar / (exp((((mu + edonor) - ec) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + ev) - mu) / kbt)) + 1.0d0))
    else
        tmp = (ndchar / (exp((((edonor + vef) - ec) / kbt)) + 1.0d0)) + (nachar / (exp((((ev + vef) + eaccept) / 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 ((mu <= -9.2e+29) || !(mu <= 1.9e+192)) {
		tmp = (NdChar / (Math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	} else {
		tmp = (NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (mu <= -9.2e+29) or not (mu <= 1.9e+192):
		tmp = (NdChar / (math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + Ev) - mu) / KbT)) + 1.0))
	else:
		tmp = (NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((mu <= -9.2e+29) || !(mu <= 1.9e+192))
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + EDonor) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)) + 1.0)));
	else
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + Vef) + EAccept) / 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 ((mu <= -9.2e+29) || ~((mu <= 1.9e+192)))
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	else
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[mu, -9.2e+29], N[Not[LessEqual[mu, 1.9e+192]], $MachinePrecision]], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / 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], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if mu < -9.2000000000000004e29 or 1.9e192 < mu

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   5. Applied rewrites 93.3%

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

   if -9.2000000000000004e29 < mu < 1.9e192

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -9.2 \cdot 10^{+29} \lor \neg \left(mu \leq 1.9 \cdot 10^{+192}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Ev + Vef\right) + EAccept\\ \mathbf{if}\;mu \leq -6 \cdot 10^{+81} \lor \neg \left(mu \leq 1.08 \cdot 10^{+193}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{t\_0 - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{t\_0}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Ev Vef) EAccept)))
   (if (or (<= mu -6e+81) (not (<= mu 1.08e+193)))
     (+
      (/ NdChar (+ 1.0 (exp (/ mu KbT))))
      (/ NaChar (+ 1.0 (exp (/ (- t_0 mu) KbT)))))
     (+
      (/ NdChar (+ (exp (/ (- (+ EDonor Vef) Ec) KbT)) 1.0))
      (/ NaChar (+ (exp (/ t_0 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 t_0 = (Ev + Vef) + EAccept;
	double tmp;
	if ((mu <= -6e+81) || !(mu <= 1.08e+193)) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp(((t_0 - mu) / KbT))));
	} else {
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((t_0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (ev + vef) + eaccept
    if ((mu <= (-6d+81)) .or. (.not. (mu <= 1.08d+193))) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp(((t_0 - mu) / kbt))))
    else
        tmp = (ndchar / (exp((((edonor + vef) - ec) / kbt)) + 1.0d0)) + (nachar / (exp((t_0 / 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 t_0 = (Ev + Vef) + EAccept;
	double tmp;
	if ((mu <= -6e+81) || !(mu <= 1.08e+193)) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp(((t_0 - mu) / KbT))));
	} else {
		tmp = (NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp((t_0 / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (Ev + Vef) + EAccept
	tmp = 0
	if (mu <= -6e+81) or not (mu <= 1.08e+193):
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp(((t_0 - mu) / KbT))))
	else:
		tmp = (NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp((t_0 / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Ev + Vef) + EAccept)
	tmp = 0.0
	if ((mu <= -6e+81) || !(mu <= 1.08e+193))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(t_0 - mu) / KbT)))));
	else
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(t_0 / KbT)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (Ev + Vef) + EAccept;
	tmp = 0.0;
	if ((mu <= -6e+81) || ~((mu <= 1.08e+193)))
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp(((t_0 - mu) / KbT))));
	else
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((t_0 / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision]}, If[Or[LessEqual[mu, -6e+81], N[Not[LessEqual[mu, 1.08e+193]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(t$95$0 - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(t$95$0 / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if mu < -5.99999999999999995e81 or 1.07999999999999996e193 < mu

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around inf

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

      1. lower-/.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if -5.99999999999999995e81 < mu < 1.07999999999999996e193

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 98.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -6 \cdot 10^{+81} \lor \neg \left(mu \leq 1.08 \cdot 10^{+193}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -4.5 \cdot 10^{+127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 10^{-162}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 1.08 \cdot 10^{+114}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
          (/ NaChar (+ (exp (/ (+ (+ Ev Vef) EAccept) KbT)) 1.0)))))
   (if (<= Vef -4.5e+127)
     t_0
     (if (<= Vef 1e-162)
       (+
        (/ NdChar (+ (exp (/ (- EDonor Ec) KbT)) 1.0))
        (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))
       (if (<= Vef 1.08e+114)
         (+
          (/ NdChar (+ (exp (/ (- (+ mu EDonor) Ec) KbT)) 1.0))
          (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp((Vef / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	double tmp;
	if (Vef <= -4.5e+127) {
		tmp = t_0;
	} else if (Vef <= 1e-162) {
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (Vef <= 1.08e+114) {
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp((vef / kbt)) + 1.0d0)) + (nachar / (exp((((ev + vef) + eaccept) / kbt)) + 1.0d0))
    if (vef <= (-4.5d+127)) then
        tmp = t_0
    else if (vef <= 1d-162) then
        tmp = (ndchar / (exp(((edonor - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    else if (vef <= 1.08d+114) then
        tmp = (ndchar / (exp((((mu + edonor) - ec) / kbt)) + 1.0d0)) + (nachar / (exp((eaccept / kbt)) + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp((Vef / KbT)) + 1.0)) + (NaChar / (Math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	double tmp;
	if (Vef <= -4.5e+127) {
		tmp = t_0;
	} else if (Vef <= 1e-162) {
		tmp = (NdChar / (Math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (Vef <= 1.08e+114) {
		tmp = (NdChar / (Math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp((Vef / KbT)) + 1.0)) + (NaChar / (math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0))
	tmp = 0
	if Vef <= -4.5e+127:
		tmp = t_0
	elif Vef <= 1e-162:
		tmp = (NdChar / (math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	elif Vef <= 1.08e+114:
		tmp = (NdChar / (math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp((EAccept / KbT)) + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + Vef) + EAccept) / KbT)) + 1.0)))
	tmp = 0.0
	if (Vef <= -4.5e+127)
		tmp = t_0;
	elseif (Vef <= 1e-162)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	elseif (Vef <= 1.08e+114)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + EDonor) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp((Vef / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	tmp = 0.0;
	if (Vef <= -4.5e+127)
		tmp = t_0;
	elseif (Vef <= 1e-162)
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	elseif (Vef <= 1.08e+114)
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((EAccept / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -4.5e+127], t$95$0, If[LessEqual[Vef, 1e-162], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 1.08e+114], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -4.50000000000000034e127 or 1.08000000000000004e114 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 82.5%

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

   if -4.50000000000000034e127 < Vef < 9.99999999999999954e-163

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in Vef around 0

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

   8. Applied rewrites 82.1%

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

   if 9.99999999999999954e-163 < Vef < 1.08000000000000004e114

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 80.9%

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

Alternative 14: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -4.5 \cdot 10^{+127} \lor \neg \left(Vef \leq 1.08 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= Vef -4.5e+127) (not (<= Vef 1.08e+114)))
   (+
    (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
    (/ NaChar (+ (exp (/ (+ (+ Ev Vef) EAccept) KbT)) 1.0)))
   (+
    (/ NdChar (+ (exp (/ (- EDonor Ec) KbT)) 1.0))
    (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((Vef <= -4.5e+127) || !(Vef <= 1.08e+114)) {
		tmp = (NdChar / (exp((Vef / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	} else {
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((vef <= (-4.5d+127)) .or. (.not. (vef <= 1.08d+114))) then
        tmp = (ndchar / (exp((vef / kbt)) + 1.0d0)) + (nachar / (exp((((ev + vef) + eaccept) / kbt)) + 1.0d0))
    else
        tmp = (ndchar / (exp(((edonor - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((Vef <= -4.5e+127) || !(Vef <= 1.08e+114)) {
		tmp = (NdChar / (Math.exp((Vef / KbT)) + 1.0)) + (NaChar / (Math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	} else {
		tmp = (NdChar / (Math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (Vef <= -4.5e+127) or not (Vef <= 1.08e+114):
		tmp = (NdChar / (math.exp((Vef / KbT)) + 1.0)) + (NaChar / (math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0))
	else:
		tmp = (NdChar / (math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((Vef <= -4.5e+127) || !(Vef <= 1.08e+114))
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + Vef) + EAccept) / KbT)) + 1.0)));
	else
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((Vef <= -4.5e+127) || ~((Vef <= 1.08e+114)))
		tmp = (NdChar / (exp((Vef / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	else
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[Vef, -4.5e+127], N[Not[LessEqual[Vef, 1.08e+114]], $MachinePrecision]], N[(N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Vef < -4.50000000000000034e127 or 1.08000000000000004e114 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 82.5%

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

   if -4.50000000000000034e127 < Vef < 1.08000000000000004e114

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in Vef around 0

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

   8. Applied rewrites 79.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -4.5 \cdot 10^{+127} \lor \neg \left(Vef \leq 1.08 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq 4.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 1.25 \cdot 10^{+199}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT 4.5e+22)
   (/ NaChar (+ (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) 1.0))
   (if (<= KbT 1.25e+199)
     (/ NdChar (+ (exp (/ (- (+ EDonor Vef) Ec) KbT)) 1.0))
     (+ (* 0.5 NdChar) (/ NaChar (+ 1.0 (exp (/ (- (+ Ev Vef) 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 <= 4.5e+22) {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else if (KbT <= 1.25e+199) {
		tmp = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((((Ev + Vef) - 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 <= 4.5d+22) then
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) + 1.0d0)
    else if (kbt <= 1.25d+199) then
        tmp = ndchar / (exp((((edonor + vef) - ec) / kbt)) + 1.0d0)
    else
        tmp = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp((((ev + vef) - 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 <= 4.5e+22) {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else if (KbT <= 1.25e+199) {
		tmp = NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp((((Ev + Vef) - mu) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= 4.5e+22:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0)
	elif KbT <= 1.25e+199:
		tmp = NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)
	else:
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + math.exp((((Ev + Vef) - mu) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= 4.5e+22)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) + 1.0));
	elseif (KbT <= 1.25e+199)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) + 1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Ev + Vef) - 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 <= 4.5e+22)
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	elseif (KbT <= 1.25e+199)
		tmp = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0);
	else
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, 4.5e+22], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.25e+199], N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Ev + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < 4.4999999999999998e22

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 4.4999999999999998e22 < KbT < 1.25e199

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in NdChar around inf

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

   11. Applied rewrites 72.2%

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

   if 1.25e199 < KbT

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in EAccept around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 87.9

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

   8. Applied rewrites 87.9%

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

Alternative 16: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq 4.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 4.3 \cdot 10^{+226}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT 4.5e+22)
   (/ NaChar (+ (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) 1.0))
   (if (<= KbT 4.3e+226)
     (/ NdChar (+ (exp (/ (- (+ EDonor Vef) Ec) KbT)) 1.0))
     (+ (* 0.5 NdChar) (/ NaChar (+ 1.0 (exp (/ Ev 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 <= 4.5e+22) {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else if (KbT <= 4.3e+226) {
		tmp = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((Ev / 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 <= 4.5d+22) then
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) + 1.0d0)
    else if (kbt <= 4.3d+226) then
        tmp = ndchar / (exp((((edonor + vef) - ec) / kbt)) + 1.0d0)
    else
        tmp = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp((ev / 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 <= 4.5e+22) {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else if (KbT <= 4.3e+226) {
		tmp = NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= 4.5e+22:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0)
	elif KbT <= 4.3e+226:
		tmp = NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)
	else:
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= 4.5e+22)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) + 1.0));
	elseif (KbT <= 4.3e+226)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) + 1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= 4.5e+22)
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	elseif (KbT <= 4.3e+226)
		tmp = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0);
	else
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((Ev / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, 4.5e+22], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.3e+226], N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < 4.4999999999999998e22

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 4.4999999999999998e22 < KbT < 4.29999999999999987e226

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 43.9%

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

   9. Taylor expanded in NdChar around inf

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

   11. Applied rewrites 70.2%

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

   if 4.29999999999999987e226 < KbT

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in Ev around inf

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

      1. lower-/.f64 88.6

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

   8. Applied rewrites 88.6%

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

Alternative 17: 62.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -4.6 \cdot 10^{-93} \lor \neg \left(NdChar \leq 4.1 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + Vef\right)}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -4.6e-93) (not (<= NdChar 4.1e+136)))
   (/ NdChar (+ (exp (/ (- (+ EDonor Vef) Ec) KbT)) 1.0))
   (/ NaChar (+ (exp (/ (+ EAccept (+ Ev Vef)) 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 <= -4.6e-93) || !(NdChar <= 4.1e+136)) {
		tmp = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp(((EAccept + (Ev + Vef)) / 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 <= (-4.6d-93)) .or. (.not. (ndchar <= 4.1d+136))) then
        tmp = ndchar / (exp((((edonor + vef) - ec) / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp(((eaccept + (ev + vef)) / 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 <= -4.6e-93) || !(NdChar <= 4.1e+136)) {
		tmp = NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + Vef)) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -4.6e-93) or not (NdChar <= 4.1e+136):
		tmp = NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp(((EAccept + (Ev + Vef)) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -4.6e-93) || !(NdChar <= 4.1e+136))
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Vef)) / 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 <= -4.6e-93) || ~((NdChar <= 4.1e+136)))
		tmp = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0);
	else
		tmp = NaChar / (exp(((EAccept + (Ev + Vef)) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -4.6e-93], N[Not[LessEqual[NdChar, 4.1e+136]], $MachinePrecision]], N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -4.5999999999999996e-93 or 4.0999999999999998e136 < NdChar

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in NdChar around inf

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

   11. Applied rewrites 61.0%

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

   if -4.5999999999999996e-93 < NdChar < 4.0999999999999998e136

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 72.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -4.6 \cdot 10^{-93} \lor \neg \left(NdChar \leq 4.1 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + Vef\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.55 \cdot 10^{+241}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{-Ec}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 1.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + Vef\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -2.55e+241)
   (/ NdChar (+ (exp (/ (- Ec) KbT)) 1.0))
   (if (<= NdChar 1.5e+138)
     (/ NaChar (+ (exp (/ (+ EAccept (+ Ev Vef)) KbT)) 1.0))
     (/ NdChar (+ (exp (/ Vef 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 <= -2.55e+241) {
		tmp = NdChar / (exp((-Ec / KbT)) + 1.0);
	} else if (NdChar <= 1.5e+138) {
		tmp = NaChar / (exp(((EAccept + (Ev + Vef)) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (exp((Vef / 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 <= (-2.55d+241)) then
        tmp = ndchar / (exp((-ec / kbt)) + 1.0d0)
    else if (ndchar <= 1.5d+138) then
        tmp = nachar / (exp(((eaccept + (ev + vef)) / kbt)) + 1.0d0)
    else
        tmp = ndchar / (exp((vef / 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 <= -2.55e+241) {
		tmp = NdChar / (Math.exp((-Ec / KbT)) + 1.0);
	} else if (NdChar <= 1.5e+138) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + Vef)) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -2.55e+241:
		tmp = NdChar / (math.exp((-Ec / KbT)) + 1.0)
	elif NdChar <= 1.5e+138:
		tmp = NaChar / (math.exp(((EAccept + (Ev + Vef)) / KbT)) + 1.0)
	else:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -2.55e+241)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(-Ec) / KbT)) + 1.0));
	elseif (NdChar <= 1.5e+138)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Vef)) / KbT)) + 1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / 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 <= -2.55e+241)
		tmp = NdChar / (exp((-Ec / KbT)) + 1.0);
	elseif (NdChar <= 1.5e+138)
		tmp = NaChar / (exp(((EAccept + (Ev + Vef)) / KbT)) + 1.0);
	else
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -2.55e+241], N[(NdChar / N[(N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.5e+138], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -2.5500000000000001e241

   1. Initial program 99.9%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 27.5%

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

   9. Taylor expanded in NdChar around inf

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

   11. Applied rewrites 70.1%

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

   12. Taylor expanded in Ec around inf

       \[\leadsto \frac{NdChar}{e^{-1 \cdot \frac{Ec}{KbT}} + 1} \]
   13. Step-by-step derivation

   14. Applied rewrites 64.4%

       \[\leadsto \frac{NdChar}{e^{\frac{-Ec}{KbT}} + 1} \]

   if -2.5500000000000001e241 < NdChar < 1.50000000000000005e138

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 65.8%

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

   if 1.50000000000000005e138 < NdChar

   1. Initial program 99.9%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in NdChar around inf

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

   11. Applied rewrites 66.7%

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

   12. Taylor expanded in Vef around inf

       \[\leadsto \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} \]
   13. Step-by-step derivation

   14. Applied rewrites 55.2%

       \[\leadsto \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 49.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -2.9 \cdot 10^{+203} \lor \neg \left(Vef \leq 2.5 \cdot 10^{+97}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= Vef -2.9e+203) (not (<= Vef 2.5e+97)))
   (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
   (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((Vef <= -2.9e+203) || !(Vef <= 2.5e+97)) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((vef <= (-2.9d+203)) .or. (.not. (vef <= 2.5d+97))) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((Vef <= -2.9e+203) || !(Vef <= 2.5e+97)) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (Vef <= -2.9e+203) or not (Vef <= 2.5e+97):
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((Vef <= -2.9e+203) || !(Vef <= 2.5e+97))
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((Vef <= -2.9e+203) || ~((Vef <= 2.5e+97)))
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[Vef, -2.9e+203], N[Not[LessEqual[Vef, 2.5e+97]], $MachinePrecision]], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Vef < -2.90000000000000011e203 or 2.49999999999999999e97 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in Vef around inf

      \[\leadsto \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.4%

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

   if -2.90000000000000011e203 < Vef < 2.49999999999999999e97

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 55.3%

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

   9. Taylor expanded in Vef around 0

      \[\leadsto \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.6%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -2.9 \cdot 10^{+203} \lor \neg \left(Vef \leq 2.5 \cdot 10^{+97}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 39.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq 3.6 \cdot 10^{+207}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept 3.6e+207)
   (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
   (/ NdChar (+ (exp (/ EDonor 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 (EAccept <= 3.6e+207) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NdChar / (exp((EDonor / 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 (eaccept <= 3.6d+207) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = ndchar / (exp((edonor / 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 (EAccept <= 3.6e+207) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EAccept <= 3.6e+207:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EAccept <= 3.6e+207)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / 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 (EAccept <= 3.6e+207)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EAccept, 3.6e+207], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if EAccept < 3.60000000000000014e207

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in Vef around inf

      \[\leadsto \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 43.6%

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

   if 3.60000000000000014e207 < EAccept

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 53.9%

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

   9. Taylor expanded in NdChar around inf

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

   11. Applied rewrites 55.1%

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

   12. Taylor expanded in EDonor around inf

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

   14. Applied rewrites 59.3%

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

Alternative 21: 40.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq 7.5 \cdot 10^{+209}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{2 + \frac{EAccept}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT 7.5e+209)
   (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
   (+ (* 0.5 NdChar) (/ NaChar (+ 2.0 (/ EAccept KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= 7.5e+209) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (2.0 + (EAccept / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= 7.5d+209) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = (0.5d0 * ndchar) + (nachar / (2.0d0 + (eaccept / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= 7.5e+209) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (2.0 + (EAccept / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= 7.5e+209:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = (0.5 * NdChar) + (NaChar / (2.0 + (EAccept / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= 7.5e+209)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(2.0 + Float64(EAccept / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= 7.5e+209)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = (0.5 * NdChar) + (NaChar / (2.0 + (EAccept / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, 7.5e+209], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < 7.50000000000000055e209

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 59.3%

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

   9. Taylor expanded in Vef around inf

      \[\leadsto \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 40.6%

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

   if 7.50000000000000055e209 < KbT

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in KbT around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-add N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 77.4

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in EAccept around inf

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

   11. Applied rewrites 74.2%

       \[\leadsto 0.5 \cdot NdChar + \frac{NaChar}{2 + \frac{EAccept}{\color{blue}{KbT}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 22.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -3.9 \cdot 10^{-69} \lor \neg \left(NaChar \leq 90000000000\right):\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -3.9e-69) (not (<= NaChar 90000000000.0)))
   (* 0.5 NaChar)
   (* 0.5 NdChar)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -3.9e-69) || !(NaChar <= 90000000000.0)) {
		tmp = 0.5 * NaChar;
	} else {
		tmp = 0.5 * NdChar;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((nachar <= (-3.9d-69)) .or. (.not. (nachar <= 90000000000.0d0))) then
        tmp = 0.5d0 * nachar
    else
        tmp = 0.5d0 * ndchar
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -3.9e-69) || !(NaChar <= 90000000000.0)) {
		tmp = 0.5 * NaChar;
	} else {
		tmp = 0.5 * NdChar;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -3.9e-69) or not (NaChar <= 90000000000.0):
		tmp = 0.5 * NaChar
	else:
		tmp = 0.5 * NdChar
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -3.9e-69) || !(NaChar <= 90000000000.0))
		tmp = Float64(0.5 * NaChar);
	else
		tmp = Float64(0.5 * NdChar);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -3.9e-69) || ~((NaChar <= 90000000000.0)))
		tmp = 0.5 * NaChar;
	else
		tmp = 0.5 * NdChar;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -3.9e-69], N[Not[LessEqual[NaChar, 90000000000.0]], $MachinePrecision]], N[(0.5 * NaChar), $MachinePrecision], N[(0.5 * NdChar), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -3.89999999999999981e-69 or 9e10 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 27.7%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   if -3.89999999999999981e-69 < NaChar < 9e10

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 27.7

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

   5. Applied rewrites 27.7%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 11.2%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

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

   11. Applied rewrites 25.0%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -3.9 \cdot 10^{-69} \lor \neg \left(NaChar \leq 90000000000\right):\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 28.3% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(NaChar + NdChar\right) \end{array} \]

FPCore

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

C

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

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 * (nachar + ndchar)
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 * (NaChar + NdChar);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 29.8

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

5. Applied rewrites 29.8%

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

Alternative 24: 17.9% accurate, 46.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot NaChar \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* 0.5 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 * 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 * 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 * NaChar;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 29.8

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

5. Applied rewrites 29.8%

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

6. Taylor expanded in NdChar around 0

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

8. Applied rewrites 21.1%

   \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(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))))))