Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 27.5s

Alternatives: 26

Speedup: 1.0×

• could not determine a ground truth

  1. branch4004 = -4.731037900914356e+25
  2. NdChar = -9.102150007445332e+25
  3. Ec = 5.773191736867147e-146
  4. Vef = -1.3818497751357656e-187
  5. EDonor = 1.3789975419955305e+126
  6. mu = 2.2258259893865442e-35
  7. KbT = -1.1059863656070945e-57
  8. NaChar = 1.9022842387019195e-233
  9. Ev = 1.5422688797790144e-229
  10. EAccept = 1.3598198776812141e+39

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}}\right)\right) + 1} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NdChar (+ (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)) 1.0))
  (/
   NaChar
   (+ (expm1 (log1p (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)))) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (expm1(log1p(exp((((Ev + EAccept) + (Vef - mu)) / KbT)))) + 1.0));
}

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.expm1(Math.log1p(Math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)))) + 1.0));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / (math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.expm1(math.log1p(math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)))) + 1.0))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(expm1(log1p(exp(Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)))) + 1.0)))
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(Exp[N[Log[1 + N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\right)\right)}} \]

   2. log1p-define 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\right)}\right)} \]

   3. log1p-expm1-u 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\right)\right)\right)}\right)} \]

   4. log1p-define 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\right)}\right)\right)\right)} \]

   5. expm1-log1p-u 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\right)\right)} \]

   6. div-inv 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\left(Vef + \left(Ev + \left(EAccept - mu\right)\right)\right) \cdot \frac{1}{KbT}}}\right)\right)} \]

   7. div-inv 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\right)\right)} \]

   8. +-commutative 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\color{blue}{\left(Ev + \left(EAccept - mu\right)\right) + Vef}}{KbT}}\right)\right)} \]

   9. associate-+r- 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\color{blue}{\left(\left(Ev + EAccept\right) - mu\right)} + Vef}{KbT}}\right)\right)} \]

   10. associate-+l- 100.0%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\color{blue}{\left(Ev + EAccept\right) - \left(mu - Vef\right)}}{KbT}}\right)\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left(Ev + EAccept\right) - \left(mu - Vef\right)}{KbT}}\right)\right)}} \]

7. Final simplification 100.0%

   \[\leadsto \frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}}\right)\right) + 1} \]
8. Add Preprocessing

Alternative 2: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ t_1 := \frac{NaChar}{t\_0 + 1}\\ t_2 := t\_1 + \frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{if}\;mu \leq -1.25 \cdot 10^{+81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;mu \leq -4.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{elif}\;mu \leq -8 \cdot 10^{-226}:\\ \;\;\;\;t\_1 + \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 2.9 \cdot 10^{+18}:\\ \;\;\;\;t\_1 + \frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))
        (t_1 (/ NaChar (+ t_0 1.0)))
        (t_2 (+ t_1 (/ NdChar (+ (exp (/ mu KbT)) 1.0)))))
   (if (<= mu -1.25e+81)
     t_2
     (if (<= mu -4.2e-109)
       (- (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (/ NaChar (- -1.0 t_0)))
       (if (<= mu -8e-226)
         (+ t_1 (/ NdChar (+ (exp (/ Vef KbT)) 1.0)))
         (if (<= mu 2.9e+18)
           (+ t_1 (/ NdChar (+ (exp (/ Ec (- KbT))) 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(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = NaChar / (t_0 + 1.0);
	double t_2 = t_1 + (NdChar / (exp((mu / KbT)) + 1.0));
	double tmp;
	if (mu <= -1.25e+81) {
		tmp = t_2;
	} else if (mu <= -4.2e-109) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0));
	} else if (mu <= -8e-226) {
		tmp = t_1 + (NdChar / (exp((Vef / KbT)) + 1.0));
	} else if (mu <= 2.9e+18) {
		tmp = t_1 + (NdChar / (exp((Ec / -KbT)) + 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) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    t_1 = nachar / (t_0 + 1.0d0)
    t_2 = t_1 + (ndchar / (exp((mu / kbt)) + 1.0d0))
    if (mu <= (-1.25d+81)) then
        tmp = t_2
    else if (mu <= (-4.2d-109)) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) - (nachar / ((-1.0d0) - t_0))
    else if (mu <= (-8d-226)) then
        tmp = t_1 + (ndchar / (exp((vef / kbt)) + 1.0d0))
    else if (mu <= 2.9d+18) then
        tmp = t_1 + (ndchar / (exp((ec / -kbt)) + 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(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = NaChar / (t_0 + 1.0);
	double t_2 = t_1 + (NdChar / (Math.exp((mu / KbT)) + 1.0));
	double tmp;
	if (mu <= -1.25e+81) {
		tmp = t_2;
	} else if (mu <= -4.2e-109) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0));
	} else if (mu <= -8e-226) {
		tmp = t_1 + (NdChar / (Math.exp((Vef / KbT)) + 1.0));
	} else if (mu <= 2.9e+18) {
		tmp = t_1 + (NdChar / (Math.exp((Ec / -KbT)) + 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	t_1 = NaChar / (t_0 + 1.0)
	t_2 = t_1 + (NdChar / (math.exp((mu / KbT)) + 1.0))
	tmp = 0
	if mu <= -1.25e+81:
		tmp = t_2
	elif mu <= -4.2e-109:
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0))
	elif mu <= -8e-226:
		tmp = t_1 + (NdChar / (math.exp((Vef / KbT)) + 1.0))
	elif mu <= 2.9e+18:
		tmp = t_1 + (NdChar / (math.exp((Ec / -KbT)) + 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(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))
	t_1 = Float64(NaChar / Float64(t_0 + 1.0))
	t_2 = Float64(t_1 + Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0)))
	tmp = 0.0
	if (mu <= -1.25e+81)
		tmp = t_2;
	elseif (mu <= -4.2e-109)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) - Float64(NaChar / Float64(-1.0 - t_0)));
	elseif (mu <= -8e-226)
		tmp = Float64(t_1 + Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0)));
	elseif (mu <= 2.9e+18)
		tmp = Float64(t_1 + Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 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(((Vef + (Ev + (EAccept - mu))) / KbT));
	t_1 = NaChar / (t_0 + 1.0);
	t_2 = t_1 + (NdChar / (exp((mu / KbT)) + 1.0));
	tmp = 0.0;
	if (mu <= -1.25e+81)
		tmp = t_2;
	elseif (mu <= -4.2e-109)
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0));
	elseif (mu <= -8e-226)
		tmp = t_1 + (NdChar / (exp((Vef / KbT)) + 1.0));
	elseif (mu <= 2.9e+18)
		tmp = t_1 + (NdChar / (exp((Ec / -KbT)) + 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[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -1.25e+81], t$95$2, If[LessEqual[mu, -4.2e-109], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, -8e-226], N[(t$95$1 + N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2.9e+18], N[(t$95$1 + N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -1.25e81 or 2.9e18 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 88.8%

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

   if -1.25e81 < mu < -4.19999999999999992e-109

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 77.9%

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

   if -4.19999999999999992e-109 < mu < -7.99999999999999937e-226

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 85.2%

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

   if -7.99999999999999937e-226 < mu < 2.9e18

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. mul-1-neg 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq -4.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;mu \leq -8 \cdot 10^{-226}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 2.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef + \left(Ev - mu\right)}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -2.5 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 7 \cdot 10^{-71}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 3.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{elif}\;Vef \leq 2.15 \cdot 10^{+158}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
          (/ NaChar (+ (exp (/ (+ Vef (- Ev mu)) KbT)) 1.0)))))
   (if (<= Vef -2.5e+116)
     t_0
     (if (<= Vef 7e-71)
       (-
        (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
        (/ NaChar (- -1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
       (if (<= Vef 3.6e+18)
         (+
          (/ NdChar (+ (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)) 1.0))
          (/
           NaChar
           (-
            (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Vef KbT) (/ Ev KbT)))
            (/ mu KbT))))
         (if (<= Vef 2.15e+158)
           (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
           t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp((Vef / KbT)) + 1.0)) + (NaChar / (exp(((Vef + (Ev - mu)) / KbT)) + 1.0));
	double tmp;
	if (Vef <= -2.5e+116) {
		tmp = t_0;
	} else if (Vef <= 7e-71) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else if (Vef <= 3.6e+18) {
		tmp = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else if (Vef <= 2.15e+158) {
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp((vef / kbt)) + 1.0d0)) + (nachar / (exp(((vef + (ev - mu)) / kbt)) + 1.0d0))
    if (vef <= (-2.5d+116)) then
        tmp = t_0
    else if (vef <= 7d-71) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) - (nachar / ((-1.0d0) - exp(((vef + (ev + (eaccept - mu))) / kbt))))
    else if (vef <= 3.6d+18) then
        tmp = (ndchar / (exp(((edonor + (mu + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (((2.0d0 + (eaccept / kbt)) + ((vef / kbt) + (ev / kbt))) - (mu / kbt)))
    else if (vef <= 2.15d+158) then
        tmp = nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp((Vef / KbT)) + 1.0)) + (NaChar / (Math.exp(((Vef + (Ev - mu)) / KbT)) + 1.0));
	double tmp;
	if (Vef <= -2.5e+116) {
		tmp = t_0;
	} else if (Vef <= 7e-71) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else if (Vef <= 3.6e+18) {
		tmp = (NdChar / (Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else if (Vef <= 2.15e+158) {
		tmp = NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp((Vef / KbT)) + 1.0)) + (NaChar / (math.exp(((Vef + (Ev - mu)) / KbT)) + 1.0))
	tmp = 0
	if Vef <= -2.5e+116:
		tmp = t_0
	elif Vef <= 7e-71:
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))))
	elif Vef <= 3.6e+18:
		tmp = (NdChar / (math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)))
	elif Vef <= 2.15e+158:
		tmp = NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(Ev - mu)) / KbT)) + 1.0)))
	tmp = 0.0
	if (Vef <= -2.5e+116)
		tmp = t_0;
	elseif (Vef <= 7e-71)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))));
	elseif (Vef <= 3.6e+18)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) - Float64(mu / KbT))));
	elseif (Vef <= 2.15e+158)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp((Vef / KbT)) + 1.0)) + (NaChar / (exp(((Vef + (Ev - mu)) / KbT)) + 1.0));
	tmp = 0.0;
	if (Vef <= -2.5e+116)
		tmp = t_0;
	elseif (Vef <= 7e-71)
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	elseif (Vef <= 3.6e+18)
		tmp = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	elseif (Vef <= 2.15e+158)
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(Vef + N[(Ev - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -2.5e+116], t$95$0, If[LessEqual[Vef, 7e-71], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 3.6e+18], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 2.15e+158], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -2.50000000000000013e116 or 2.15e158 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 80.4%

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

   6. Taylor expanded in EAccept around 0 79.1%

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

      1. +-commutative 79.1%

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

      2. +-commutative 79.1%

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

      3. associate--l+ 79.1%

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

   8. Simplified 79.1%

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

   if -2.50000000000000013e116 < Vef < 6.9999999999999998e-71

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 78.5%

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

   if 6.9999999999999998e-71 < Vef < 3.6e18

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in KbT around inf 91.6%

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

      1. associate-+r+ 91.6%

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

   7. Simplified 91.6%

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

   if 3.6e18 < Vef < 2.15e158

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 52.8%

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

      1. associate-*r/ 52.8%

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

      2. mul-1-neg 52.8%

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

   7. Simplified 52.8%

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

   8. Taylor expanded in NdChar around 0 69.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -2.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef + \left(Ev - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 7 \cdot 10^{-71}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 3.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{elif}\;Vef \leq 2.15 \cdot 10^{+158}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef + \left(Ev - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{Vef + \left(Ev - mu\right)}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + t\_0\\ \mathbf{if}\;EDonor \leq -5.8 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;EDonor \leq 2050000000000:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + t\_0\\ \mathbf{elif}\;EDonor \leq 1.16 \cdot 10^{+126}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ Vef (- Ev mu)) KbT)) 1.0)))
        (t_1 (+ (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) t_0)))
   (if (<= EDonor -5.8e-52)
     t_1
     (if (<= EDonor 2050000000000.0)
       (+ (/ NdChar (+ (exp (/ Vef KbT)) 1.0)) t_0)
       (if (<= EDonor 1.16e+126)
         (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) 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 / (exp(((Vef + (Ev - mu)) / KbT)) + 1.0);
	double t_1 = (NdChar / (exp((EDonor / KbT)) + 1.0)) + t_0;
	double tmp;
	if (EDonor <= -5.8e-52) {
		tmp = t_1;
	} else if (EDonor <= 2050000000000.0) {
		tmp = (NdChar / (exp((Vef / KbT)) + 1.0)) + t_0;
	} else if (EDonor <= 1.16e+126) {
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / 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 / (exp(((vef + (ev - mu)) / kbt)) + 1.0d0)
    t_1 = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + t_0
    if (edonor <= (-5.8d-52)) then
        tmp = t_1
    else if (edonor <= 2050000000000.0d0) then
        tmp = (ndchar / (exp((vef / kbt)) + 1.0d0)) + t_0
    else if (edonor <= 1.16d+126) then
        tmp = nachar / (exp((((eaccept + (vef + ev)) - mu) / 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 / (Math.exp(((Vef + (Ev - mu)) / KbT)) + 1.0);
	double t_1 = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + t_0;
	double tmp;
	if (EDonor <= -5.8e-52) {
		tmp = t_1;
	} else if (EDonor <= 2050000000000.0) {
		tmp = (NdChar / (Math.exp((Vef / KbT)) + 1.0)) + t_0;
	} else if (EDonor <= 1.16e+126) {
		tmp = NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp(((Vef + (Ev - mu)) / KbT)) + 1.0)
	t_1 = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + t_0
	tmp = 0
	if EDonor <= -5.8e-52:
		tmp = t_1
	elif EDonor <= 2050000000000.0:
		tmp = (NdChar / (math.exp((Vef / KbT)) + 1.0)) + t_0
	elif EDonor <= 1.16e+126:
		tmp = NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / 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(exp(Float64(Float64(Vef + Float64(Ev - mu)) / KbT)) + 1.0))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + t_0)
	tmp = 0.0
	if (EDonor <= -5.8e-52)
		tmp = t_1;
	elseif (EDonor <= 2050000000000.0)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) + t_0);
	elseif (EDonor <= 1.16e+126)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / 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 / (exp(((Vef + (Ev - mu)) / KbT)) + 1.0);
	t_1 = (NdChar / (exp((EDonor / KbT)) + 1.0)) + t_0;
	tmp = 0.0;
	if (EDonor <= -5.8e-52)
		tmp = t_1;
	elseif (EDonor <= 2050000000000.0)
		tmp = (NdChar / (exp((Vef / KbT)) + 1.0)) + t_0;
	elseif (EDonor <= 1.16e+126)
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / 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[(N[Exp[N[(N[(Vef + N[(Ev - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[EDonor, -5.8e-52], t$95$1, If[LessEqual[EDonor, 2050000000000.0], N[(N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[EDonor, 1.16e+126], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if EDonor < -5.8000000000000003e-52 or 1.15999999999999997e126 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 83.6%

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

   6. Taylor expanded in EAccept around 0 76.7%

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

      1. +-commutative 53.9%

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

      2. +-commutative 53.9%

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

      3. associate--l+ 53.9%

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

   8. Simplified 76.7%

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

   if -5.8000000000000003e-52 < EDonor < 2.05e12

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Vef around inf 74.7%

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

   6. Taylor expanded in EAccept around 0 70.3%

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

      1. +-commutative 70.3%

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

      2. +-commutative 70.3%

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

      3. associate--l+ 70.3%

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

   8. Simplified 70.3%

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

   if 2.05e12 < EDonor < 1.15999999999999997e126

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 67.7%

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

      1. associate-*r/ 67.7%

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

      2. mul-1-neg 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in NdChar around 0 94.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -5.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef + \left(Ev - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 2050000000000:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef + \left(Ev - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 1.16 \cdot 10^{+126}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef + \left(Ev - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{if}\;NaChar \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{\left(\left(\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) + 1\right) - \frac{Ec}{KbT}\right) + 1}\\ \mathbf{elif}\;NaChar \leq 9.5 \cdot 10^{-183}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 3.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef + \left(Ev - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)) 1.0))
          (/
           NaChar
           (-
            (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Vef KbT) (/ Ev KbT)))
            (/ mu KbT))))))
   (if (<= NaChar -4e-5)
     (+
      (/ NaChar (+ (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)) 1.0))
      (/
       NdChar
       (+
        (- (+ (+ (/ EDonor KbT) (+ (/ mu KbT) (/ Vef KbT))) 1.0) (/ Ec KbT))
        1.0)))
     (if (<= NaChar 9.5e-183)
       t_0
       (if (<= NaChar 3.5e-69)
         (+
          (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
          (/ NaChar (+ (exp (/ (+ Vef (- Ev mu)) KbT)) 1.0)))
         (if (<= NaChar 5.2e-6)
           t_0
           (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	double tmp;
	if (NaChar <= -4e-5) {
		tmp = (NaChar / (exp(((Vef + (Ev + (EAccept - mu))) / KbT)) + 1.0)) + (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0));
	} else if (NaChar <= 9.5e-183) {
		tmp = t_0;
	} else if (NaChar <= 3.5e-69) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / (exp(((Vef + (Ev - mu)) / KbT)) + 1.0));
	} else if (NaChar <= 5.2e-6) {
		tmp = t_0;
	} else {
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp(((edonor + (mu + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (((2.0d0 + (eaccept / kbt)) + ((vef / kbt) + (ev / kbt))) - (mu / kbt)))
    if (nachar <= (-4d-5)) then
        tmp = (nachar / (exp(((vef + (ev + (eaccept - mu))) / kbt)) + 1.0d0)) + (ndchar / (((((edonor / kbt) + ((mu / kbt) + (vef / kbt))) + 1.0d0) - (ec / kbt)) + 1.0d0))
    else if (nachar <= 9.5d-183) then
        tmp = t_0
    else if (nachar <= 3.5d-69) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar / (exp(((vef + (ev - mu)) / kbt)) + 1.0d0))
    else if (nachar <= 5.2d-6) then
        tmp = t_0
    else
        tmp = nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	double tmp;
	if (NaChar <= -4e-5) {
		tmp = (NaChar / (Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)) + 1.0)) + (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0));
	} else if (NaChar <= 9.5e-183) {
		tmp = t_0;
	} else if (NaChar <= 3.5e-69) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar / (Math.exp(((Vef + (Ev - mu)) / KbT)) + 1.0));
	} else if (NaChar <= 5.2e-6) {
		tmp = t_0;
	} else {
		tmp = NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)))
	tmp = 0
	if NaChar <= -4e-5:
		tmp = (NaChar / (math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)) + 1.0)) + (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0))
	elif NaChar <= 9.5e-183:
		tmp = t_0
	elif NaChar <= 3.5e-69:
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar / (math.exp(((Vef + (Ev - mu)) / KbT)) + 1.0))
	elif NaChar <= 5.2e-6:
		tmp = t_0
	else:
		tmp = NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) - Float64(mu / KbT))))
	tmp = 0.0
	if (NaChar <= -4e-5)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)) + 1.0)) + Float64(NdChar / Float64(Float64(Float64(Float64(Float64(EDonor / KbT) + Float64(Float64(mu / KbT) + Float64(Vef / KbT))) + 1.0) - Float64(Ec / KbT)) + 1.0)));
	elseif (NaChar <= 9.5e-183)
		tmp = t_0;
	elseif (NaChar <= 3.5e-69)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(Ev - mu)) / KbT)) + 1.0)));
	elseif (NaChar <= 5.2e-6)
		tmp = t_0;
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	tmp = 0.0;
	if (NaChar <= -4e-5)
		tmp = (NaChar / (exp(((Vef + (Ev + (EAccept - mu))) / KbT)) + 1.0)) + (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0));
	elseif (NaChar <= 9.5e-183)
		tmp = t_0;
	elseif (NaChar <= 3.5e-69)
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / (exp(((Vef + (Ev - mu)) / KbT)) + 1.0));
	elseif (NaChar <= 5.2e-6)
		tmp = t_0;
	else
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / 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[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -4e-5], N[(N[(NaChar / N[(N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[(N[(N[(N[(EDonor / KbT), $MachinePrecision] + N[(N[(mu / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 9.5e-183], t$95$0, If[LessEqual[NaChar, 3.5e-69], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(Vef + N[(Ev - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 5.2e-6], t$95$0, N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -4.00000000000000033e-5

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 69.5%

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

   if -4.00000000000000033e-5 < NaChar < 9.5000000000000008e-183 or 3.5000000000000001e-69 < NaChar < 5.20000000000000019e-6

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf 74.3%

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

      1. associate-+r+ 74.3%

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

   7. Simplified 74.3%

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

   if 9.5000000000000008e-183 < NaChar < 3.5000000000000001e-69

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 73.7%

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

   6. Taylor expanded in EAccept around 0 69.7%

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

      1. +-commutative 65.5%

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

      2. +-commutative 65.5%

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

      3. associate--l+ 65.5%

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

   8. Simplified 69.7%

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

   if 5.20000000000000019e-6 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 79.4%

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

      1. associate-*r/ 79.4%

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

      2. mul-1-neg 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in NdChar around 0 76.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{\left(\left(\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) + 1\right) - \frac{Ec}{KbT}\right) + 1}\\ \mathbf{elif}\;NaChar \leq 9.5 \cdot 10^{-183}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{elif}\;NaChar \leq 3.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef + \left(Ev - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ \mathbf{if}\;EDonor \leq -1.04 \cdot 10^{+65} \lor \neg \left(EDonor \leq 1.2 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_0 + 1} + \frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
   (if (or (<= EDonor -1.04e+65) (not (<= EDonor 1.2e+137)))
     (- (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (/ NaChar (- -1.0 t_0)))
     (+ (/ NaChar (+ t_0 1.0)) (/ NdChar (+ (exp (/ mu KbT)) 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if ((EDonor <= -1.04e+65) || !(EDonor <= 1.2e+137)) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = (NaChar / (t_0 + 1.0)) + (NdChar / (exp((mu / KbT)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    if ((edonor <= (-1.04d+65)) .or. (.not. (edonor <= 1.2d+137))) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) - (nachar / ((-1.0d0) - t_0))
    else
        tmp = (nachar / (t_0 + 1.0d0)) + (ndchar / (exp((mu / kbt)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if ((EDonor <= -1.04e+65) || !(EDonor <= 1.2e+137)) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = (NaChar / (t_0 + 1.0)) + (NdChar / (Math.exp((mu / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	tmp = 0
	if (EDonor <= -1.04e+65) or not (EDonor <= 1.2e+137):
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0))
	else:
		tmp = (NaChar / (t_0 + 1.0)) + (NdChar / (math.exp((mu / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))
	tmp = 0.0
	if ((EDonor <= -1.04e+65) || !(EDonor <= 1.2e+137))
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) - Float64(NaChar / Float64(-1.0 - t_0)));
	else
		tmp = Float64(Float64(NaChar / Float64(t_0 + 1.0)) + Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	tmp = 0.0;
	if ((EDonor <= -1.04e+65) || ~((EDonor <= 1.2e+137)))
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0));
	else
		tmp = (NaChar / (t_0 + 1.0)) + (NdChar / (exp((mu / KbT)) + 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[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[EDonor, -1.04e+65], N[Not[LessEqual[EDonor, 1.2e+137]], $MachinePrecision]], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if EDonor < -1.03999999999999999e65 or 1.19999999999999992e137 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 85.8%

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

   if -1.03999999999999999e65 < EDonor < 1.19999999999999992e137

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in mu around inf 77.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -1.04 \cdot 10^{+65} \lor \neg \left(EDonor \leq 1.2 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ \mathbf{if}\;EDonor \leq -2.55 \cdot 10^{-51} \lor \neg \left(EDonor \leq 2 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_0 + 1} + \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
   (if (or (<= EDonor -2.55e-51) (not (<= EDonor 2e+138)))
     (- (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (/ NaChar (- -1.0 t_0)))
     (+ (/ NaChar (+ t_0 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 t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if ((EDonor <= -2.55e-51) || !(EDonor <= 2e+138)) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = (NaChar / (t_0 + 1.0)) + (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) :: t_0
    real(8) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    if ((edonor <= (-2.55d-51)) .or. (.not. (edonor <= 2d+138))) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) - (nachar / ((-1.0d0) - t_0))
    else
        tmp = (nachar / (t_0 + 1.0d0)) + (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 t_0 = Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if ((EDonor <= -2.55e-51) || !(EDonor <= 2e+138)) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = (NaChar / (t_0 + 1.0)) + (NdChar / (Math.exp((Vef / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	tmp = 0
	if (EDonor <= -2.55e-51) or not (EDonor <= 2e+138):
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0))
	else:
		tmp = (NaChar / (t_0 + 1.0)) + (NdChar / (math.exp((Vef / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))
	tmp = 0.0
	if ((EDonor <= -2.55e-51) || !(EDonor <= 2e+138))
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) - Float64(NaChar / Float64(-1.0 - t_0)));
	else
		tmp = Float64(Float64(NaChar / Float64(t_0 + 1.0)) + 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)
	t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	tmp = 0.0;
	if ((EDonor <= -2.55e-51) || ~((EDonor <= 2e+138)))
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) - (NaChar / (-1.0 - t_0));
	else
		tmp = (NaChar / (t_0 + 1.0)) + (NdChar / (exp((Vef / KbT)) + 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[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[EDonor, -2.55e-51], N[Not[LessEqual[EDonor, 2e+138]], $MachinePrecision]], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if EDonor < -2.5499999999999999e-51 or 2.0000000000000001e138 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 84.6%

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

   if -2.5499999999999999e-51 < EDonor < 2.0000000000000001e138

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Vef around inf 74.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -2.55 \cdot 10^{-51} \lor \neg \left(EDonor \leq 2 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 9: 61.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{if}\;NdChar \leq -4.6 \cdot 10^{+179}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\frac{NaChar}{\left(\mathsf{expm1}\left(\frac{Ev + \left(Vef + \left(EAccept - mu\right)\right)}{KbT}\right) + 1\right) + 1}\\ \mathbf{elif}\;NdChar \leq 6.8 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 0.0015:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 1.9 \cdot 10^{+90} \lor \neg \left(NdChar \leq 1.9 \cdot 10^{+250}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)) 1.0))
          (/ NaChar 2.0))))
   (if (<= NdChar -4.6e+179)
     t_0
     (if (<= NdChar 4e-55)
       (/ NaChar (+ (+ (expm1 (/ (+ Ev (+ Vef (- EAccept mu))) KbT)) 1.0) 1.0))
       (if (<= NdChar 6.8e-22)
         t_0
         (if (<= NdChar 0.0015)
           (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
           (if (or (<= NdChar 1.9e+90) (not (<= NdChar 1.9e+250)))
             t_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 t_0 = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0);
	double tmp;
	if (NdChar <= -4.6e+179) {
		tmp = t_0;
	} else if (NdChar <= 4e-55) {
		tmp = NaChar / ((expm1(((Ev + (Vef + (EAccept - mu))) / KbT)) + 1.0) + 1.0);
	} else if (NdChar <= 6.8e-22) {
		tmp = t_0;
	} else if (NdChar <= 0.0015) {
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	} else if ((NdChar <= 1.9e+90) || !(NdChar <= 1.9e+250)) {
		tmp = t_0;
	} else {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	}
	return tmp;
}

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0);
	double tmp;
	if (NdChar <= -4.6e+179) {
		tmp = t_0;
	} else if (NdChar <= 4e-55) {
		tmp = NaChar / ((Math.expm1(((Ev + (Vef + (EAccept - mu))) / KbT)) + 1.0) + 1.0);
	} else if (NdChar <= 6.8e-22) {
		tmp = t_0;
	} else if (NdChar <= 0.0015) {
		tmp = NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	} else if ((NdChar <= 1.9e+90) || !(NdChar <= 1.9e+250)) {
		tmp = t_0;
	} else {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0)
	tmp = 0
	if NdChar <= -4.6e+179:
		tmp = t_0
	elif NdChar <= 4e-55:
		tmp = NaChar / ((math.expm1(((Ev + (Vef + (EAccept - mu))) / KbT)) + 1.0) + 1.0)
	elif NdChar <= 6.8e-22:
		tmp = t_0
	elif NdChar <= 0.0015:
		tmp = NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)
	elif (NdChar <= 1.9e+90) or not (NdChar <= 1.9e+250):
		tmp = t_0
	else:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / 2.0))
	tmp = 0.0
	if (NdChar <= -4.6e+179)
		tmp = t_0;
	elseif (NdChar <= 4e-55)
		tmp = Float64(NaChar / Float64(Float64(expm1(Float64(Float64(Ev + Float64(Vef + Float64(EAccept - mu))) / KbT)) + 1.0) + 1.0));
	elseif (NdChar <= 6.8e-22)
		tmp = t_0;
	elseif (NdChar <= 0.0015)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0));
	elseif ((NdChar <= 1.9e+90) || !(NdChar <= 1.9e+250))
		tmp = t_0;
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -4.6e+179], t$95$0, If[LessEqual[NdChar, 4e-55], N[(NaChar / N[(N[(N[(Exp[N[(N[(Ev + N[(Vef + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]] - 1), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 6.8e-22], t$95$0, If[LessEqual[NdChar, 0.0015], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[NdChar, 1.9e+90], N[Not[LessEqual[NdChar, 1.9e+250]], $MachinePrecision]], t$95$0, N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -4.59999999999999988e179 or 3.99999999999999998e-55 < NdChar < 6.7999999999999997e-22 or 0.0015 < NdChar < 1.9000000000000001e90 or 1.89999999999999985e250 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 70.6%

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

   if -4.59999999999999988e179 < NdChar < 3.99999999999999998e-55

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 72.8%

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

      1. associate-*r/ 72.8%

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

      2. mul-1-neg 72.8%

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

   7. Simplified 72.8%

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

   8. Taylor expanded in NdChar around 0 70.3%

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

      1. expm1-log1p-u 70.3%

         \[\leadsto \frac{NaChar}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)\right)}} \]

      2. log1p-define 70.3%

         \[\leadsto \frac{NaChar}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)} \]

      3. expm1-undefine 70.3%

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

      4. add-exp-log 70.3%

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

      5. associate--l+ 70.3%

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

   10. Applied egg-rr 70.3%

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

       1. associate--l+ 70.3%

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

       2. expm1-define 70.3%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \color{blue}{\mathsf{expm1}\left(\frac{EAccept + \left(\left(Ev + Vef\right) - mu\right)}{KbT}\right)}\right)} \]

       3. +-commutative 70.3%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\left(\left(Ev + Vef\right) - mu\right) + EAccept}}{KbT}\right)\right)} \]

       4. *-lft-identity 70.3%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\left(\left(Ev + Vef\right) - mu\right) + \color{blue}{1 \cdot EAccept}}{KbT}\right)\right)} \]

       5. metadata-eval 70.3%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\left(\left(Ev + Vef\right) - mu\right) + \color{blue}{\left(--1\right)} \cdot EAccept}{KbT}\right)\right)} \]

       6. cancel-sign-sub-inv 70.3%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\left(\left(Ev + Vef\right) - mu\right) - -1 \cdot EAccept}}{KbT}\right)\right)} \]

       7. associate--r+ 70.3%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\left(Ev + Vef\right) - \left(mu + -1 \cdot EAccept\right)}}{KbT}\right)\right)} \]

       8. associate--l+ 70.3%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{Ev + \left(Vef - \left(mu + -1 \cdot EAccept\right)\right)}}{KbT}\right)\right)} \]

       9. mul-1-neg 70.3%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{Ev + \left(Vef - \left(mu + \color{blue}{\left(-EAccept\right)}\right)\right)}{KbT}\right)\right)} \]

       10. unsub-neg 70.3%

           \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{Ev + \left(Vef - \color{blue}{\left(mu - EAccept\right)}\right)}{KbT}\right)\right)} \]

   12. Simplified 70.3%

       \[\leadsto \frac{NaChar}{1 + \color{blue}{\left(1 + \mathsf{expm1}\left(\frac{Ev + \left(Vef - \left(mu - EAccept\right)\right)}{KbT}\right)\right)}} \]

   if 6.7999999999999997e-22 < NdChar < 0.0015

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. mul-1-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in NdChar around 0 78.5%

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

   if 1.9000000000000001e90 < NdChar < 1.89999999999999985e250

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 78.0%

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

   6. Taylor expanded in KbT around inf 43.1%

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

      1. associate-+r+ 54.0%

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

   8. Simplified 43.1%

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

   9. Taylor expanded in Vef around inf 40.3%

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

   10. Taylor expanded in NdChar around inf 66.6%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -4.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;NdChar \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\frac{NaChar}{\left(\mathsf{expm1}\left(\frac{Ev + \left(Vef + \left(EAccept - mu\right)\right)}{KbT}\right) + 1\right) + 1}\\ \mathbf{elif}\;NdChar \leq 6.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;NdChar \leq 0.0015:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 1.9 \cdot 10^{+90} \lor \neg \left(NdChar \leq 1.9 \cdot 10^{+250}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{\left(\left(\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) + 1\right) - \frac{Ec}{KbT}\right) + 1}\\ \mathbf{elif}\;NaChar \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -4.6e-5)
   (+
    (/ NaChar (+ (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)) 1.0))
    (/
     NdChar
     (+
      (- (+ (+ (/ EDonor KbT) (+ (/ mu KbT) (/ Vef KbT))) 1.0) (/ Ec KbT))
      1.0)))
   (if (<= NaChar 5.5e-8)
     (+
      (/ NdChar (+ (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)) 1.0))
      (/
       NaChar
       (- (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Vef KbT) (/ Ev KbT))) (/ mu KbT))))
     (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -4.6e-5) {
		tmp = (NaChar / (exp(((Vef + (Ev + (EAccept - mu))) / KbT)) + 1.0)) + (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0));
	} else if (NaChar <= 5.5e-8) {
		tmp = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else {
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-4.6d-5)) then
        tmp = (nachar / (exp(((vef + (ev + (eaccept - mu))) / kbt)) + 1.0d0)) + (ndchar / (((((edonor / kbt) + ((mu / kbt) + (vef / kbt))) + 1.0d0) - (ec / kbt)) + 1.0d0))
    else if (nachar <= 5.5d-8) then
        tmp = (ndchar / (exp(((edonor + (mu + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (((2.0d0 + (eaccept / kbt)) + ((vef / kbt) + (ev / kbt))) - (mu / kbt)))
    else
        tmp = nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -4.6e-5) {
		tmp = (NaChar / (Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)) + 1.0)) + (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0));
	} else if (NaChar <= 5.5e-8) {
		tmp = (NdChar / (Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else {
		tmp = NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -4.6e-5:
		tmp = (NaChar / (math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)) + 1.0)) + (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0))
	elif NaChar <= 5.5e-8:
		tmp = (NdChar / (math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)))
	else:
		tmp = NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -4.6e-5)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)) + 1.0)) + Float64(NdChar / Float64(Float64(Float64(Float64(Float64(EDonor / KbT) + Float64(Float64(mu / KbT) + Float64(Vef / KbT))) + 1.0) - Float64(Ec / KbT)) + 1.0)));
	elseif (NaChar <= 5.5e-8)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) - Float64(mu / KbT))));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -4.6e-5)
		tmp = (NaChar / (exp(((Vef + (Ev + (EAccept - mu))) / KbT)) + 1.0)) + (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0));
	elseif (NaChar <= 5.5e-8)
		tmp = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	else
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -4.6e-5], N[(N[(NaChar / N[(N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[(N[(N[(N[(EDonor / KbT), $MachinePrecision] + N[(N[(mu / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 5.5e-8], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -4.6e-5

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 69.5%

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

   if -4.6e-5 < NaChar < 5.5000000000000003e-8

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf 70.1%

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

      1. associate-+r+ 70.1%

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

   7. Simplified 70.1%

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

   if 5.5000000000000003e-8 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 79.4%

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

      1. associate-*r/ 79.4%

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

      2. mul-1-neg 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in NdChar around 0 76.8%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{\left(\left(\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) + 1\right) - \frac{Ec}{KbT}\right) + 1}\\ \mathbf{elif}\;NaChar \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -5.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 1} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -5.6e+42)
   (-
    (/ NdChar (+ (/ Vef KbT) 1.0))
    (/ NaChar (- -1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
   (if (<= NaChar 9e-6)
     (+
      (/ NdChar (+ (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)) 1.0))
      (/
       NaChar
       (- (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Vef KbT) (/ Ev KbT))) (/ mu KbT))))
     (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -5.6e+42) {
		tmp = (NdChar / ((Vef / KbT) + 1.0)) - (NaChar / (-1.0 - exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else if (NaChar <= 9e-6) {
		tmp = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else {
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-5.6d+42)) then
        tmp = (ndchar / ((vef / kbt) + 1.0d0)) - (nachar / ((-1.0d0) - exp(((vef + (ev + (eaccept - mu))) / kbt))))
    else if (nachar <= 9d-6) then
        tmp = (ndchar / (exp(((edonor + (mu + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (((2.0d0 + (eaccept / kbt)) + ((vef / kbt) + (ev / kbt))) - (mu / kbt)))
    else
        tmp = nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -5.6e+42) {
		tmp = (NdChar / ((Vef / KbT) + 1.0)) - (NaChar / (-1.0 - Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else if (NaChar <= 9e-6) {
		tmp = (NdChar / (Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else {
		tmp = NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -5.6e+42:
		tmp = (NdChar / ((Vef / KbT) + 1.0)) - (NaChar / (-1.0 - math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))))
	elif NaChar <= 9e-6:
		tmp = (NdChar / (math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)))
	else:
		tmp = NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -5.6e+42)
		tmp = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))));
	elseif (NaChar <= 9e-6)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) - Float64(mu / KbT))));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -5.6e+42)
		tmp = (NdChar / ((Vef / KbT) + 1.0)) - (NaChar / (-1.0 - exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	elseif (NaChar <= 9e-6)
		tmp = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	else
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -5.6e+42], N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 9e-6], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -5.5999999999999999e42

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 70.0%

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

   6. Taylor expanded in Vef around inf 71.9%

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

   if -5.5999999999999999e42 < NaChar < 9.00000000000000023e-6

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf 68.8%

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

      1. associate-+r+ 68.8%

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

   7. Simplified 68.8%

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

   if 9.00000000000000023e-6 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 79.4%

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

      1. associate-*r/ 79.4%

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

      2. mul-1-neg 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in NdChar around 0 76.8%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 1} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{mu}{-KbT}} + 1}\\ t_1 := e^{\frac{Vef}{KbT}} + 1\\ t_2 := \frac{NaChar}{t\_1}\\ \mathbf{if}\;NaChar \leq -1.5 \cdot 10^{+157}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NaChar \leq -3 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) + 1\right) - \frac{Ec}{KbT}\right) + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;NaChar \leq -2.75 \cdot 10^{-39}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -9.5 \cdot 10^{-51}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NaChar \leq 1900000000:\\ \;\;\;\;\frac{NdChar}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ mu (- KbT))) 1.0)))
        (t_1 (+ (exp (/ Vef KbT)) 1.0))
        (t_2 (/ NaChar t_1)))
   (if (<= NaChar -1.5e+157)
     t_2
     (if (<= NaChar -3e+24)
       t_0
       (if (<= NaChar -6.5e-6)
         (+
          (/
           NdChar
           (+
            (-
             (+ (+ (/ EDonor KbT) (+ (/ mu KbT) (/ Vef KbT))) 1.0)
             (/ Ec KbT))
            1.0))
          (/ NaChar 2.0))
         (if (<= NaChar -2.75e-39)
           (/ NdChar (+ (exp (/ Ec (- KbT))) 1.0))
           (if (<= NaChar -9.5e-51)
             t_2
             (if (<= NaChar 1900000000.0) (/ NdChar t_1) t_0))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((mu / -KbT)) + 1.0);
	double t_1 = exp((Vef / KbT)) + 1.0;
	double t_2 = NaChar / t_1;
	double tmp;
	if (NaChar <= -1.5e+157) {
		tmp = t_2;
	} else if (NaChar <= -3e+24) {
		tmp = t_0;
	} else if (NaChar <= -6.5e-6) {
		tmp = (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0)) + (NaChar / 2.0);
	} else if (NaChar <= -2.75e-39) {
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	} else if (NaChar <= -9.5e-51) {
		tmp = t_2;
	} else if (NaChar <= 1900000000.0) {
		tmp = NdChar / t_1;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (exp((mu / -kbt)) + 1.0d0)
    t_1 = exp((vef / kbt)) + 1.0d0
    t_2 = nachar / t_1
    if (nachar <= (-1.5d+157)) then
        tmp = t_2
    else if (nachar <= (-3d+24)) then
        tmp = t_0
    else if (nachar <= (-6.5d-6)) then
        tmp = (ndchar / (((((edonor / kbt) + ((mu / kbt) + (vef / kbt))) + 1.0d0) - (ec / kbt)) + 1.0d0)) + (nachar / 2.0d0)
    else if (nachar <= (-2.75d-39)) then
        tmp = ndchar / (exp((ec / -kbt)) + 1.0d0)
    else if (nachar <= (-9.5d-51)) then
        tmp = t_2
    else if (nachar <= 1900000000.0d0) then
        tmp = ndchar / t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((mu / -KbT)) + 1.0);
	double t_1 = Math.exp((Vef / KbT)) + 1.0;
	double t_2 = NaChar / t_1;
	double tmp;
	if (NaChar <= -1.5e+157) {
		tmp = t_2;
	} else if (NaChar <= -3e+24) {
		tmp = t_0;
	} else if (NaChar <= -6.5e-6) {
		tmp = (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0)) + (NaChar / 2.0);
	} else if (NaChar <= -2.75e-39) {
		tmp = NdChar / (Math.exp((Ec / -KbT)) + 1.0);
	} else if (NaChar <= -9.5e-51) {
		tmp = t_2;
	} else if (NaChar <= 1900000000.0) {
		tmp = NdChar / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((mu / -KbT)) + 1.0)
	t_1 = math.exp((Vef / KbT)) + 1.0
	t_2 = NaChar / t_1
	tmp = 0
	if NaChar <= -1.5e+157:
		tmp = t_2
	elif NaChar <= -3e+24:
		tmp = t_0
	elif NaChar <= -6.5e-6:
		tmp = (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0)) + (NaChar / 2.0)
	elif NaChar <= -2.75e-39:
		tmp = NdChar / (math.exp((Ec / -KbT)) + 1.0)
	elif NaChar <= -9.5e-51:
		tmp = t_2
	elif NaChar <= 1900000000.0:
		tmp = NdChar / t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(mu / Float64(-KbT))) + 1.0))
	t_1 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	t_2 = Float64(NaChar / t_1)
	tmp = 0.0
	if (NaChar <= -1.5e+157)
		tmp = t_2;
	elseif (NaChar <= -3e+24)
		tmp = t_0;
	elseif (NaChar <= -6.5e-6)
		tmp = Float64(Float64(NdChar / Float64(Float64(Float64(Float64(Float64(EDonor / KbT) + Float64(Float64(mu / KbT) + Float64(Vef / KbT))) + 1.0) - Float64(Ec / KbT)) + 1.0)) + Float64(NaChar / 2.0));
	elseif (NaChar <= -2.75e-39)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	elseif (NaChar <= -9.5e-51)
		tmp = t_2;
	elseif (NaChar <= 1900000000.0)
		tmp = Float64(NdChar / t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((mu / -KbT)) + 1.0);
	t_1 = exp((Vef / KbT)) + 1.0;
	t_2 = NaChar / t_1;
	tmp = 0.0;
	if (NaChar <= -1.5e+157)
		tmp = t_2;
	elseif (NaChar <= -3e+24)
		tmp = t_0;
	elseif (NaChar <= -6.5e-6)
		tmp = (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0)) + (NaChar / 2.0);
	elseif (NaChar <= -2.75e-39)
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	elseif (NaChar <= -9.5e-51)
		tmp = t_2;
	elseif (NaChar <= 1900000000.0)
		tmp = NdChar / t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / t$95$1), $MachinePrecision]}, If[LessEqual[NaChar, -1.5e+157], t$95$2, If[LessEqual[NaChar, -3e+24], t$95$0, If[LessEqual[NaChar, -6.5e-6], N[(N[(NdChar / N[(N[(N[(N[(N[(EDonor / KbT), $MachinePrecision] + N[(N[(mu / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, -2.75e-39], N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, -9.5e-51], t$95$2, If[LessEqual[NaChar, 1900000000.0], N[(NdChar / t$95$1), $MachinePrecision], t$95$0]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NaChar < -1.50000000000000005e157 or -2.75000000000000009e-39 < NaChar < -9.4999999999999998e-51

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 69.3%

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

      1. associate-*r/ 69.3%

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

      2. mul-1-neg 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in NdChar around 0 70.8%

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

   9. Taylor expanded in Vef around inf 62.1%

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

   10. Taylor expanded in mu around 0 59.7%

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

   if -1.50000000000000005e157 < NaChar < -2.99999999999999995e24 or 1.9e9 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 79.9%

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

      1. associate-*r/ 79.9%

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

      2. mul-1-neg 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in NdChar around 0 76.8%

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

   9. Taylor expanded in mu around inf 49.6%

      \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{-1 \cdot mu}}{KbT}}} \]
   10. Step-by-step derivation

       1. neg-mul-1 49.6%

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

   11. Simplified 49.6%

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

   if -2.99999999999999995e24 < NaChar < -6.4999999999999996e-6

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 61.8%

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

   6. Taylor expanded in KbT around inf 54.7%

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

   if -6.4999999999999996e-6 < NaChar < -2.75000000000000009e-39

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 78.3%

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

      1. associate-*r/ 78.3%

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

      2. mul-1-neg 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in NdChar around inf 78.3%

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

      1. mul-1-neg 78.3%

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

      2. distribute-frac-neg 78.3%

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

   10. Simplified 78.3%

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

   if -9.4999999999999998e-51 < NaChar < 1.9e9

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Vef around inf 62.7%

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

   6. Taylor expanded in KbT around inf 46.7%

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

      1. associate-+r+ 68.2%

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

   8. Simplified 46.7%

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

   9. Taylor expanded in Vef around inf 35.4%

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

   10. Taylor expanded in NdChar around inf 52.2%

       \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -3 \cdot 10^{+24}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{mu}{-KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) + 1\right) - \frac{Ec}{KbT}\right) + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;NaChar \leq -2.75 \cdot 10^{-39}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -9.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 1900000000:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{mu}{-KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;mu \leq -2.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 1} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;mu \leq -5.1 \cdot 10^{-85}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq -4.2 \cdot 10^{-180}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 4.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(\mathsf{expm1}\left(\frac{Ev + \left(Vef + \left(EAccept - mu\right)\right)}{KbT}\right) + 1\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= mu -2.7e+85)
   (-
    (/ NdChar (+ (/ Vef KbT) 1.0))
    (/ NaChar (- -1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
   (if (<= mu -5.1e-85)
     (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
     (if (<= mu -4.2e-180)
       (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
       (if (<= mu 4.2e-210)
         (+
          (/ NdChar (+ (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)) 1.0))
          (/ NaChar 2.0))
         (/
          NaChar
          (+ (+ (expm1 (/ (+ Ev (+ Vef (- EAccept mu))) KbT)) 1.0) 1.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= -2.7e+85) {
		tmp = (NdChar / ((Vef / KbT) + 1.0)) - (NaChar / (-1.0 - exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else if (mu <= -5.1e-85) {
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	} else if (mu <= -4.2e-180) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else if (mu <= 4.2e-210) {
		tmp = (NdChar / (exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0);
	} else {
		tmp = NaChar / ((expm1(((Ev + (Vef + (EAccept - mu))) / KbT)) + 1.0) + 1.0);
	}
	return tmp;
}

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 <= -2.7e+85) {
		tmp = (NdChar / ((Vef / KbT) + 1.0)) - (NaChar / (-1.0 - Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else if (mu <= -5.1e-85) {
		tmp = NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	} else if (mu <= -4.2e-180) {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	} else if (mu <= 4.2e-210) {
		tmp = (NdChar / (Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0);
	} else {
		tmp = NaChar / ((Math.expm1(((Ev + (Vef + (EAccept - mu))) / KbT)) + 1.0) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if mu <= -2.7e+85:
		tmp = (NdChar / ((Vef / KbT) + 1.0)) - (NaChar / (-1.0 - math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))))
	elif mu <= -5.1e-85:
		tmp = NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)
	elif mu <= -4.2e-180:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	elif mu <= 4.2e-210:
		tmp = (NdChar / (math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0)
	else:
		tmp = NaChar / ((math.expm1(((Ev + (Vef + (EAccept - mu))) / KbT)) + 1.0) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (mu <= -2.7e+85)
		tmp = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))));
	elseif (mu <= -5.1e-85)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0));
	elseif (mu <= -4.2e-180)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (mu <= 4.2e-210)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / 2.0));
	else
		tmp = Float64(NaChar / Float64(Float64(expm1(Float64(Float64(Ev + Float64(Vef + Float64(EAccept - mu))) / KbT)) + 1.0) + 1.0));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[mu, -2.7e+85], N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, -5.1e-85], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, -4.2e-180], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 4.2e-210], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[(N[(Exp[N[(N[(Ev + N[(Vef + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]] - 1), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if mu < -2.69999999999999983e85

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 37.9%

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

   6. Taylor expanded in Vef around inf 63.2%

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

   if -2.69999999999999983e85 < mu < -5.1000000000000002e-85

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. mul-1-neg 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in NdChar around 0 74.4%

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

   if -5.1000000000000002e-85 < mu < -4.1999999999999997e-180

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 82.3%

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

   6. Taylor expanded in KbT around inf 44.7%

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

      1. associate-+r+ 57.8%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in Vef around inf 35.5%

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

   10. Taylor expanded in NdChar around inf 67.1%

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

   if -4.1999999999999997e-180 < mu < 4.20000000000000032e-210

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.5%

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

   if 4.20000000000000032e-210 < mu

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. mul-1-neg 71.4%

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

   7. Simplified 71.4%

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

   8. Taylor expanded in NdChar around 0 67.4%

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

      1. expm1-log1p-u 67.4%

         \[\leadsto \frac{NaChar}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)\right)}} \]

      2. log1p-define 67.4%

         \[\leadsto \frac{NaChar}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)} \]

      3. expm1-undefine 67.4%

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

      4. add-exp-log 67.4%

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

      5. associate--l+ 67.4%

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

   10. Applied egg-rr 67.4%

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

       1. associate--l+ 67.4%

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

       2. expm1-define 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \color{blue}{\mathsf{expm1}\left(\frac{EAccept + \left(\left(Ev + Vef\right) - mu\right)}{KbT}\right)}\right)} \]

       3. +-commutative 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\left(\left(Ev + Vef\right) - mu\right) + EAccept}}{KbT}\right)\right)} \]

       4. *-lft-identity 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\left(\left(Ev + Vef\right) - mu\right) + \color{blue}{1 \cdot EAccept}}{KbT}\right)\right)} \]

       5. metadata-eval 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\left(\left(Ev + Vef\right) - mu\right) + \color{blue}{\left(--1\right)} \cdot EAccept}{KbT}\right)\right)} \]

       6. cancel-sign-sub-inv 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\left(\left(Ev + Vef\right) - mu\right) - -1 \cdot EAccept}}{KbT}\right)\right)} \]

       7. associate--r+ 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\left(Ev + Vef\right) - \left(mu + -1 \cdot EAccept\right)}}{KbT}\right)\right)} \]

       8. associate--l+ 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{Ev + \left(Vef - \left(mu + -1 \cdot EAccept\right)\right)}}{KbT}\right)\right)} \]

       9. mul-1-neg 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{Ev + \left(Vef - \left(mu + \color{blue}{\left(-EAccept\right)}\right)\right)}{KbT}\right)\right)} \]

       10. unsub-neg 67.4%

           \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{Ev + \left(Vef - \color{blue}{\left(mu - EAccept\right)}\right)}{KbT}\right)\right)} \]

   12. Simplified 67.4%

       \[\leadsto \frac{NaChar}{1 + \color{blue}{\left(1 + \mathsf{expm1}\left(\frac{Ev + \left(Vef - \left(mu - EAccept\right)\right)}{KbT}\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 1} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;mu \leq -5.1 \cdot 10^{-85}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq -4.2 \cdot 10^{-180}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 4.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(\mathsf{expm1}\left(\frac{Ev + \left(Vef + \left(EAccept - mu\right)\right)}{KbT}\right) + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;mu \leq -2.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq -9.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 3.6 \cdot 10^{-210}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{Ec}{-KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(\mathsf{expm1}\left(\frac{Ev + \left(Vef + \left(EAccept - mu\right)\right)}{KbT}\right) + 1\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= mu -2.6e-85)
   (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
   (if (<= mu -9.5e-180)
     (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
     (if (<= mu 3.6e-210)
       (- (/ NaChar 2.0) (/ NdChar (- -1.0 (exp (/ Ec (- KbT))))))
       (/
        NaChar
        (+ (+ (expm1 (/ (+ Ev (+ Vef (- EAccept mu))) KbT)) 1.0) 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= -2.6e-85) {
		tmp = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	} else if (mu <= -9.5e-180) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else if (mu <= 3.6e-210) {
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - exp((Ec / -KbT))));
	} else {
		tmp = NaChar / ((expm1(((Ev + (Vef + (EAccept - mu))) / KbT)) + 1.0) + 1.0);
	}
	return tmp;
}

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 <= -2.6e-85) {
		tmp = NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	} else if (mu <= -9.5e-180) {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	} else if (mu <= 3.6e-210) {
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - Math.exp((Ec / -KbT))));
	} else {
		tmp = NaChar / ((Math.expm1(((Ev + (Vef + (EAccept - mu))) / KbT)) + 1.0) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if mu <= -2.6e-85:
		tmp = NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)
	elif mu <= -9.5e-180:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	elif mu <= 3.6e-210:
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - math.exp((Ec / -KbT))))
	else:
		tmp = NaChar / ((math.expm1(((Ev + (Vef + (EAccept - mu))) / KbT)) + 1.0) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (mu <= -2.6e-85)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0));
	elseif (mu <= -9.5e-180)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (mu <= 3.6e-210)
		tmp = Float64(Float64(NaChar / 2.0) - Float64(NdChar / Float64(-1.0 - exp(Float64(Ec / Float64(-KbT))))));
	else
		tmp = Float64(NaChar / Float64(Float64(expm1(Float64(Float64(Ev + Float64(Vef + Float64(EAccept - mu))) / KbT)) + 1.0) + 1.0));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[mu, -2.6e-85], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, -9.5e-180], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 3.6e-210], N[(N[(NaChar / 2.0), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[(N[(Exp[N[(N[(Ev + N[(Vef + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]] - 1), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -2.60000000000000011e-85

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 58.9%

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

      1. associate-*r/ 58.9%

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

      2. mul-1-neg 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in NdChar around 0 60.8%

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

   if -2.60000000000000011e-85 < mu < -9.49999999999999934e-180

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 82.3%

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

   6. Taylor expanded in KbT around inf 44.7%

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

      1. associate-+r+ 57.8%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in Vef around inf 35.5%

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

   10. Taylor expanded in NdChar around inf 67.1%

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

   if -9.49999999999999934e-180 < mu < 3.5999999999999999e-210

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.5%

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

   6. Taylor expanded in Ec around inf 63.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

   8. Simplified 63.3%

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

   if 3.5999999999999999e-210 < mu

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. mul-1-neg 71.4%

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

   7. Simplified 71.4%

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

   8. Taylor expanded in NdChar around 0 67.4%

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

      1. expm1-log1p-u 67.4%

         \[\leadsto \frac{NaChar}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)\right)}} \]

      2. log1p-define 67.4%

         \[\leadsto \frac{NaChar}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)} \]

      3. expm1-undefine 67.4%

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

      4. add-exp-log 67.4%

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

      5. associate--l+ 67.4%

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

   10. Applied egg-rr 67.4%

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

       1. associate--l+ 67.4%

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

       2. expm1-define 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \color{blue}{\mathsf{expm1}\left(\frac{EAccept + \left(\left(Ev + Vef\right) - mu\right)}{KbT}\right)}\right)} \]

       3. +-commutative 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\left(\left(Ev + Vef\right) - mu\right) + EAccept}}{KbT}\right)\right)} \]

       4. *-lft-identity 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\left(\left(Ev + Vef\right) - mu\right) + \color{blue}{1 \cdot EAccept}}{KbT}\right)\right)} \]

       5. metadata-eval 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\left(\left(Ev + Vef\right) - mu\right) + \color{blue}{\left(--1\right)} \cdot EAccept}{KbT}\right)\right)} \]

       6. cancel-sign-sub-inv 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\left(\left(Ev + Vef\right) - mu\right) - -1 \cdot EAccept}}{KbT}\right)\right)} \]

       7. associate--r+ 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{\left(Ev + Vef\right) - \left(mu + -1 \cdot EAccept\right)}}{KbT}\right)\right)} \]

       8. associate--l+ 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{\color{blue}{Ev + \left(Vef - \left(mu + -1 \cdot EAccept\right)\right)}}{KbT}\right)\right)} \]

       9. mul-1-neg 67.4%

          \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{Ev + \left(Vef - \left(mu + \color{blue}{\left(-EAccept\right)}\right)\right)}{KbT}\right)\right)} \]

       10. unsub-neg 67.4%

           \[\leadsto \frac{NaChar}{1 + \left(1 + \mathsf{expm1}\left(\frac{Ev + \left(Vef - \color{blue}{\left(mu - EAccept\right)}\right)}{KbT}\right)\right)} \]

   12. Simplified 67.4%

       \[\leadsto \frac{NaChar}{1 + \color{blue}{\left(1 + \mathsf{expm1}\left(\frac{Ev + \left(Vef - \left(mu - EAccept\right)\right)}{KbT}\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq -9.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 3.6 \cdot 10^{-210}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{Ec}{-KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(\mathsf{expm1}\left(\frac{Ev + \left(Vef + \left(EAccept - mu\right)\right)}{KbT}\right) + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -1.95 \cdot 10^{+123}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev - mu}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef - mu}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -1.22 \cdot 10^{-257}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;Ev \leq 4 \cdot 10^{-187}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -1.95e+123)
   (/ NaChar (+ (exp (/ (- Ev mu) KbT)) 1.0))
   (if (<= Ev -5e-161)
     (/ NaChar (+ (exp (/ (- Vef mu) KbT)) 1.0))
     (if (<= Ev -1.22e-257)
       (+ (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (/ NaChar 2.0))
       (if (<= Ev 4e-187)
         (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
         (/ NaChar (+ (exp (/ (- EAccept mu) KbT)) 1.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -1.95e+123) {
		tmp = NaChar / (exp(((Ev - mu) / KbT)) + 1.0);
	} else if (Ev <= -5e-161) {
		tmp = NaChar / (exp(((Vef - mu) / KbT)) + 1.0);
	} else if (Ev <= -1.22e-257) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	} else if (Ev <= 4e-187) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ev <= (-1.95d+123)) then
        tmp = nachar / (exp(((ev - mu) / kbt)) + 1.0d0)
    else if (ev <= (-5d-161)) then
        tmp = nachar / (exp(((vef - mu) / kbt)) + 1.0d0)
    else if (ev <= (-1.22d-257)) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar / 2.0d0)
    else if (ev <= 4d-187) then
        tmp = ndchar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp(((eaccept - mu) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -1.95e+123) {
		tmp = NaChar / (Math.exp(((Ev - mu) / KbT)) + 1.0);
	} else if (Ev <= -5e-161) {
		tmp = NaChar / (Math.exp(((Vef - mu) / KbT)) + 1.0);
	} else if (Ev <= -1.22e-257) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	} else if (Ev <= 4e-187) {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp(((EAccept - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -1.95e+123:
		tmp = NaChar / (math.exp(((Ev - mu) / KbT)) + 1.0)
	elif Ev <= -5e-161:
		tmp = NaChar / (math.exp(((Vef - mu) / KbT)) + 1.0)
	elif Ev <= -1.22e-257:
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0)
	elif Ev <= 4e-187:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp(((EAccept - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -1.95e+123)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Ev - mu) / KbT)) + 1.0));
	elseif (Ev <= -5e-161)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Vef - mu) / KbT)) + 1.0));
	elseif (Ev <= -1.22e-257)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar / 2.0));
	elseif (Ev <= 4e-187)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -1.95e+123)
		tmp = NaChar / (exp(((Ev - mu) / KbT)) + 1.0);
	elseif (Ev <= -5e-161)
		tmp = NaChar / (exp(((Vef - mu) / KbT)) + 1.0);
	elseif (Ev <= -1.22e-257)
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	elseif (Ev <= 4e-187)
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -1.95e+123], N[(NaChar / N[(N[Exp[N[(N[(Ev - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, -5e-161], N[(NaChar / N[(N[Exp[N[(N[(Vef - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, -1.22e-257], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, 4e-187], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(EAccept - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if Ev < -1.94999999999999996e123

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 56.1%

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

      1. associate-*r/ 56.1%

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

      2. mul-1-neg 56.1%

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

   7. Simplified 56.1%

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

   8. Taylor expanded in NdChar around 0 46.7%

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

   9. Taylor expanded in Ev around inf 43.3%

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

   if -1.94999999999999996e123 < Ev < -4.9999999999999999e-161

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 74.2%

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

      1. associate-*r/ 74.2%

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

      2. mul-1-neg 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in NdChar around 0 63.0%

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

   9. Taylor expanded in Vef around inf 53.3%

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

   if -4.9999999999999999e-161 < Ev < -1.22000000000000004e-257

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in KbT around inf 56.9%

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

   6. Taylor expanded in EDonor around inf 45.6%

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

   if -1.22000000000000004e-257 < Ev < 4.0000000000000001e-187

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 78.6%

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

   6. Taylor expanded in KbT around inf 44.7%

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

      1. associate-+r+ 55.5%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in Vef around inf 30.4%

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

   10. Taylor expanded in NdChar around inf 41.7%

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

   if 4.0000000000000001e-187 < Ev

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 69.3%

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

      1. associate-*r/ 69.3%

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

      2. mul-1-neg 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in NdChar around 0 62.4%

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

   9. Taylor expanded in EAccept around inf 45.1%

      \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept} - mu}{KbT}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -1.95 \cdot 10^{+123}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev - mu}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef - mu}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -1.22 \cdot 10^{-257}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;Ev \leq 4 \cdot 10^{-187}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;NaChar \leq -6 \cdot 10^{-6}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev - mu}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -3.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -6.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{elif}\;NaChar \leq 12500000000:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0)))
   (if (<= NaChar -6e-6)
     (/ NaChar (+ (exp (/ (- Ev mu) KbT)) 1.0))
     (if (<= NaChar -3.6e-39)
       (/ NdChar (+ (exp (/ Ec (- KbT))) 1.0))
       (if (<= NaChar -6.6e-54)
         (/ NaChar t_0)
         (if (<= NaChar 12500000000.0)
           (/ NdChar t_0)
           (/ NaChar (+ (exp (/ (- EAccept mu) KbT)) 1.0))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((Vef / KbT)) + 1.0;
	double tmp;
	if (NaChar <= -6e-6) {
		tmp = NaChar / (exp(((Ev - mu) / KbT)) + 1.0);
	} else if (NaChar <= -3.6e-39) {
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	} else if (NaChar <= -6.6e-54) {
		tmp = NaChar / t_0;
	} else if (NaChar <= 12500000000.0) {
		tmp = NdChar / t_0;
	} else {
		tmp = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    if (nachar <= (-6d-6)) then
        tmp = nachar / (exp(((ev - mu) / kbt)) + 1.0d0)
    else if (nachar <= (-3.6d-39)) then
        tmp = ndchar / (exp((ec / -kbt)) + 1.0d0)
    else if (nachar <= (-6.6d-54)) then
        tmp = nachar / t_0
    else if (nachar <= 12500000000.0d0) then
        tmp = ndchar / t_0
    else
        tmp = nachar / (exp(((eaccept - mu) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp((Vef / KbT)) + 1.0;
	double tmp;
	if (NaChar <= -6e-6) {
		tmp = NaChar / (Math.exp(((Ev - mu) / KbT)) + 1.0);
	} else if (NaChar <= -3.6e-39) {
		tmp = NdChar / (Math.exp((Ec / -KbT)) + 1.0);
	} else if (NaChar <= -6.6e-54) {
		tmp = NaChar / t_0;
	} else if (NaChar <= 12500000000.0) {
		tmp = NdChar / t_0;
	} else {
		tmp = NaChar / (Math.exp(((EAccept - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	tmp = 0
	if NaChar <= -6e-6:
		tmp = NaChar / (math.exp(((Ev - mu) / KbT)) + 1.0)
	elif NaChar <= -3.6e-39:
		tmp = NdChar / (math.exp((Ec / -KbT)) + 1.0)
	elif NaChar <= -6.6e-54:
		tmp = NaChar / t_0
	elif NaChar <= 12500000000.0:
		tmp = NdChar / t_0
	else:
		tmp = NaChar / (math.exp(((EAccept - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	tmp = 0.0
	if (NaChar <= -6e-6)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Ev - mu) / KbT)) + 1.0));
	elseif (NaChar <= -3.6e-39)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	elseif (NaChar <= -6.6e-54)
		tmp = Float64(NaChar / t_0);
	elseif (NaChar <= 12500000000.0)
		tmp = Float64(NdChar / t_0);
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp((Vef / KbT)) + 1.0;
	tmp = 0.0;
	if (NaChar <= -6e-6)
		tmp = NaChar / (exp(((Ev - mu) / KbT)) + 1.0);
	elseif (NaChar <= -3.6e-39)
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	elseif (NaChar <= -6.6e-54)
		tmp = NaChar / t_0;
	elseif (NaChar <= 12500000000.0)
		tmp = NdChar / t_0;
	else
		tmp = NaChar / (exp(((EAccept - mu) / 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[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[NaChar, -6e-6], N[(NaChar / N[(N[Exp[N[(N[(Ev - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, -3.6e-39], N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, -6.6e-54], N[(NaChar / t$95$0), $MachinePrecision], If[LessEqual[NaChar, 12500000000.0], N[(NdChar / t$95$0), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(EAccept - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if NaChar < -6.0000000000000002e-6

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 72.9%

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

      1. associate-*r/ 72.9%

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

      2. mul-1-neg 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in NdChar around 0 69.2%

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

   9. Taylor expanded in Ev around inf 52.8%

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

   if -6.0000000000000002e-6 < NaChar < -3.6000000000000001e-39

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 78.3%

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

      1. associate-*r/ 78.3%

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

      2. mul-1-neg 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in NdChar around inf 78.3%

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

      1. mul-1-neg 78.3%

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

      2. distribute-frac-neg 78.3%

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

   10. Simplified 78.3%

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

   if -3.6000000000000001e-39 < NaChar < -6.59999999999999986e-54

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 55.8%

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

      1. associate-*r/ 55.8%

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

      2. mul-1-neg 55.8%

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

   7. Simplified 55.8%

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

   8. Taylor expanded in NdChar around 0 75.5%

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

   9. Taylor expanded in Vef around inf 75.9%

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

   10. Taylor expanded in mu around 0 75.9%

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

   if -6.59999999999999986e-54 < NaChar < 1.25e10

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Vef around inf 62.7%

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

   6. Taylor expanded in KbT around inf 46.7%

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

      1. associate-+r+ 68.2%

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

   8. Simplified 46.7%

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

   9. Taylor expanded in Vef around inf 35.4%

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

   10. Taylor expanded in NdChar around inf 52.2%

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

   if 1.25e10 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 80.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

   7. Simplified 80.3%

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

   8. Taylor expanded in NdChar around 0 79.3%

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

   9. Taylor expanded in EAccept around inf 62.8%

      \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept} - mu}{KbT}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -6 \cdot 10^{-6}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev - mu}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -3.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -6.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 12500000000:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ t_1 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;Vef \leq -4.9 \cdot 10^{+104}:\\ \;\;\;\;\frac{NdChar}{t\_1}\\ \mathbf{elif}\;Vef \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 4.3 \cdot 10^{+20}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 2.95 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (- EAccept mu) KbT)) 1.0)))
        (t_1 (+ (exp (/ Vef KbT)) 1.0)))
   (if (<= Vef -4.9e+104)
     (/ NdChar t_1)
     (if (<= Vef 2.5e-70)
       t_0
       (if (<= Vef 4.3e+20)
         (/ NdChar (+ (exp (/ Ec (- KbT))) 1.0))
         (if (<= Vef 2.95e+109) t_0 (/ NaChar 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 / (exp(((EAccept - mu) / KbT)) + 1.0);
	double t_1 = exp((Vef / KbT)) + 1.0;
	double tmp;
	if (Vef <= -4.9e+104) {
		tmp = NdChar / t_1;
	} else if (Vef <= 2.5e-70) {
		tmp = t_0;
	} else if (Vef <= 4.3e+20) {
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	} else if (Vef <= 2.95e+109) {
		tmp = t_0;
	} else {
		tmp = NaChar / 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 / (exp(((eaccept - mu) / kbt)) + 1.0d0)
    t_1 = exp((vef / kbt)) + 1.0d0
    if (vef <= (-4.9d+104)) then
        tmp = ndchar / t_1
    else if (vef <= 2.5d-70) then
        tmp = t_0
    else if (vef <= 4.3d+20) then
        tmp = ndchar / (exp((ec / -kbt)) + 1.0d0)
    else if (vef <= 2.95d+109) then
        tmp = t_0
    else
        tmp = nachar / 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 / (Math.exp(((EAccept - mu) / KbT)) + 1.0);
	double t_1 = Math.exp((Vef / KbT)) + 1.0;
	double tmp;
	if (Vef <= -4.9e+104) {
		tmp = NdChar / t_1;
	} else if (Vef <= 2.5e-70) {
		tmp = t_0;
	} else if (Vef <= 4.3e+20) {
		tmp = NdChar / (Math.exp((Ec / -KbT)) + 1.0);
	} else if (Vef <= 2.95e+109) {
		tmp = t_0;
	} else {
		tmp = NaChar / t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp(((EAccept - mu) / KbT)) + 1.0)
	t_1 = math.exp((Vef / KbT)) + 1.0
	tmp = 0
	if Vef <= -4.9e+104:
		tmp = NdChar / t_1
	elif Vef <= 2.5e-70:
		tmp = t_0
	elif Vef <= 4.3e+20:
		tmp = NdChar / (math.exp((Ec / -KbT)) + 1.0)
	elif Vef <= 2.95e+109:
		tmp = t_0
	else:
		tmp = NaChar / t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(EAccept - mu) / KbT)) + 1.0))
	t_1 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	tmp = 0.0
	if (Vef <= -4.9e+104)
		tmp = Float64(NdChar / t_1);
	elseif (Vef <= 2.5e-70)
		tmp = t_0;
	elseif (Vef <= 4.3e+20)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	elseif (Vef <= 2.95e+109)
		tmp = t_0;
	else
		tmp = Float64(NaChar / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	t_1 = exp((Vef / KbT)) + 1.0;
	tmp = 0.0;
	if (Vef <= -4.9e+104)
		tmp = NdChar / t_1;
	elseif (Vef <= 2.5e-70)
		tmp = t_0;
	elseif (Vef <= 4.3e+20)
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	elseif (Vef <= 2.95e+109)
		tmp = t_0;
	else
		tmp = NaChar / 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[(N[Exp[N[(N[(EAccept - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[Vef, -4.9e+104], N[(NdChar / t$95$1), $MachinePrecision], If[LessEqual[Vef, 2.5e-70], t$95$0, If[LessEqual[Vef, 4.3e+20], N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 2.95e+109], t$95$0, N[(NaChar / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -4.89999999999999985e104

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 78.8%

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

   6. Taylor expanded in KbT around inf 41.1%

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

      1. associate-+r+ 56.9%

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

   8. Simplified 41.1%

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

   9. Taylor expanded in Vef around inf 42.6%

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

   10. Taylor expanded in NdChar around inf 64.2%

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

   if -4.89999999999999985e104 < Vef < 2.4999999999999999e-70 or 4.3e20 < Vef < 2.9499999999999999e109

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 76.9%

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

      1. associate-*r/ 76.9%

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

      2. mul-1-neg 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in NdChar around 0 62.3%

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

   9. Taylor expanded in EAccept around inf 52.2%

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

   if 2.4999999999999999e-70 < Vef < 4.3e20

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in Ec around inf 91.9%

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

      1. associate-*r/ 91.9%

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

      2. mul-1-neg 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in NdChar around inf 75.7%

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

      1. mul-1-neg 75.7%

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

      2. distribute-frac-neg 75.7%

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

   10. Simplified 75.7%

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

   if 2.9499999999999999e109 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 43.6%

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

      1. associate-*r/ 43.6%

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

      2. mul-1-neg 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in NdChar around 0 66.0%

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

   9. Taylor expanded in Vef around inf 63.6%

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

   10. Taylor expanded in mu around 0 61.2%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -4.9 \cdot 10^{+104}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 4.3 \cdot 10^{+20}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 2.95 \cdot 10^{+109}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{mu}{-KbT}} + 1}\\ t_1 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;NaChar \leq -2.75 \cdot 10^{+157}:\\ \;\;\;\;\frac{NaChar}{t\_1}\\ \mathbf{elif}\;NaChar \leq -2.45 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq -7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) + 1\right) - \frac{Ec}{KbT}\right) + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;NaChar \leq 10000000000:\\ \;\;\;\;\frac{NdChar}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ mu (- KbT))) 1.0)))
        (t_1 (+ (exp (/ Vef KbT)) 1.0)))
   (if (<= NaChar -2.75e+157)
     (/ NaChar t_1)
     (if (<= NaChar -2.45e+25)
       t_0
       (if (<= NaChar -7.8e-7)
         (+
          (/
           NdChar
           (+
            (-
             (+ (+ (/ EDonor KbT) (+ (/ mu KbT) (/ Vef KbT))) 1.0)
             (/ Ec KbT))
            1.0))
          (/ NaChar 2.0))
         (if (<= NaChar 10000000000.0) (/ NdChar t_1) t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((mu / -KbT)) + 1.0);
	double t_1 = exp((Vef / KbT)) + 1.0;
	double tmp;
	if (NaChar <= -2.75e+157) {
		tmp = NaChar / t_1;
	} else if (NaChar <= -2.45e+25) {
		tmp = t_0;
	} else if (NaChar <= -7.8e-7) {
		tmp = (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0)) + (NaChar / 2.0);
	} else if (NaChar <= 10000000000.0) {
		tmp = NdChar / t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = nachar / (exp((mu / -kbt)) + 1.0d0)
    t_1 = exp((vef / kbt)) + 1.0d0
    if (nachar <= (-2.75d+157)) then
        tmp = nachar / t_1
    else if (nachar <= (-2.45d+25)) then
        tmp = t_0
    else if (nachar <= (-7.8d-7)) then
        tmp = (ndchar / (((((edonor / kbt) + ((mu / kbt) + (vef / kbt))) + 1.0d0) - (ec / kbt)) + 1.0d0)) + (nachar / 2.0d0)
    else if (nachar <= 10000000000.0d0) then
        tmp = ndchar / t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((mu / -KbT)) + 1.0);
	double t_1 = Math.exp((Vef / KbT)) + 1.0;
	double tmp;
	if (NaChar <= -2.75e+157) {
		tmp = NaChar / t_1;
	} else if (NaChar <= -2.45e+25) {
		tmp = t_0;
	} else if (NaChar <= -7.8e-7) {
		tmp = (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0)) + (NaChar / 2.0);
	} else if (NaChar <= 10000000000.0) {
		tmp = NdChar / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((mu / -KbT)) + 1.0)
	t_1 = math.exp((Vef / KbT)) + 1.0
	tmp = 0
	if NaChar <= -2.75e+157:
		tmp = NaChar / t_1
	elif NaChar <= -2.45e+25:
		tmp = t_0
	elif NaChar <= -7.8e-7:
		tmp = (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0)) + (NaChar / 2.0)
	elif NaChar <= 10000000000.0:
		tmp = NdChar / t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(mu / Float64(-KbT))) + 1.0))
	t_1 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	tmp = 0.0
	if (NaChar <= -2.75e+157)
		tmp = Float64(NaChar / t_1);
	elseif (NaChar <= -2.45e+25)
		tmp = t_0;
	elseif (NaChar <= -7.8e-7)
		tmp = Float64(Float64(NdChar / Float64(Float64(Float64(Float64(Float64(EDonor / KbT) + Float64(Float64(mu / KbT) + Float64(Vef / KbT))) + 1.0) - Float64(Ec / KbT)) + 1.0)) + Float64(NaChar / 2.0));
	elseif (NaChar <= 10000000000.0)
		tmp = Float64(NdChar / t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((mu / -KbT)) + 1.0);
	t_1 = exp((Vef / KbT)) + 1.0;
	tmp = 0.0;
	if (NaChar <= -2.75e+157)
		tmp = NaChar / t_1;
	elseif (NaChar <= -2.45e+25)
		tmp = t_0;
	elseif (NaChar <= -7.8e-7)
		tmp = (NdChar / (((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) + 1.0) - (Ec / KbT)) + 1.0)) + (NaChar / 2.0);
	elseif (NaChar <= 10000000000.0)
		tmp = NdChar / t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[NaChar, -2.75e+157], N[(NaChar / t$95$1), $MachinePrecision], If[LessEqual[NaChar, -2.45e+25], t$95$0, If[LessEqual[NaChar, -7.8e-7], N[(N[(NdChar / N[(N[(N[(N[(N[(EDonor / KbT), $MachinePrecision] + N[(N[(mu / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 10000000000.0], N[(NdChar / t$95$1), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -2.7500000000000001e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 71.1%

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

      1. associate-*r/ 71.1%

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

      2. mul-1-neg 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in NdChar around 0 70.1%

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

   9. Taylor expanded in Vef around inf 60.2%

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

   10. Taylor expanded in mu around 0 57.4%

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

   if -2.7500000000000001e157 < NaChar < -2.45e25 or 1e10 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 79.9%

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

      1. associate-*r/ 79.9%

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

      2. mul-1-neg 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in NdChar around 0 76.8%

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

   9. Taylor expanded in mu around inf 49.6%

      \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{-1 \cdot mu}}{KbT}}} \]
   10. Step-by-step derivation

       1. neg-mul-1 49.6%

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

   11. Simplified 49.6%

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

   if -2.45e25 < NaChar < -7.80000000000000049e-7

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 57.9%

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

   6. Taylor expanded in KbT around inf 51.3%

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

   if -7.80000000000000049e-7 < NaChar < 1e10

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Vef around inf 62.1%

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

   6. Taylor expanded in KbT around inf 45.3%

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

      1. associate-+r+ 67.8%

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

   8. Simplified 45.3%

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

   9. Taylor expanded in Vef around inf 35.0%

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

   10. Taylor expanded in NdChar around inf 51.0%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.75 \cdot 10^{+157}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -2.45 \cdot 10^{+25}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{mu}{-KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq -7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) + 1\right) - \frac{Ec}{KbT}\right) + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;NaChar \leq 10000000000:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{mu}{-KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;mu \leq -2.6 \cdot 10^{-85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq -7 \cdot 10^{-181}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 3.9 \cdot 10^{-210}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{Ec}{-KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))))
   (if (<= mu -2.6e-85)
     t_0
     (if (<= mu -7e-181)
       (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
       (if (<= mu 3.9e-210)
         (- (/ NaChar 2.0) (/ NdChar (- -1.0 (exp (/ Ec (- KbT))))))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	double tmp;
	if (mu <= -2.6e-85) {
		tmp = t_0;
	} else if (mu <= -7e-181) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else if (mu <= 3.9e-210) {
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - exp((Ec / -KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)
    if (mu <= (-2.6d-85)) then
        tmp = t_0
    else if (mu <= (-7d-181)) then
        tmp = ndchar / (exp((vef / kbt)) + 1.0d0)
    else if (mu <= 3.9d-210) then
        tmp = (nachar / 2.0d0) - (ndchar / ((-1.0d0) - exp((ec / -kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	double tmp;
	if (mu <= -2.6e-85) {
		tmp = t_0;
	} else if (mu <= -7e-181) {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	} else if (mu <= 3.9e-210) {
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - Math.exp((Ec / -KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)
	tmp = 0
	if mu <= -2.6e-85:
		tmp = t_0
	elif mu <= -7e-181:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	elif mu <= 3.9e-210:
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - math.exp((Ec / -KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0))
	tmp = 0.0
	if (mu <= -2.6e-85)
		tmp = t_0;
	elseif (mu <= -7e-181)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (mu <= 3.9e-210)
		tmp = Float64(Float64(NaChar / 2.0) - Float64(NdChar / Float64(-1.0 - exp(Float64(Ec / Float64(-KbT))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	tmp = 0.0;
	if (mu <= -2.6e-85)
		tmp = t_0;
	elseif (mu <= -7e-181)
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	elseif (mu <= 3.9e-210)
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - exp((Ec / -KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -2.6e-85], t$95$0, If[LessEqual[mu, -7e-181], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 3.9e-210], N[(N[(NaChar / 2.0), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -2.60000000000000011e-85 or 3.8999999999999998e-210 < mu

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 66.5%

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

      1. associate-*r/ 66.5%

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

      2. mul-1-neg 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in NdChar around 0 64.8%

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

   if -2.60000000000000011e-85 < mu < -6.99999999999999993e-181

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 82.3%

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

   6. Taylor expanded in KbT around inf 44.7%

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

      1. associate-+r+ 57.8%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in Vef around inf 35.5%

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

   10. Taylor expanded in NdChar around inf 67.1%

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

   if -6.99999999999999993e-181 < mu < 3.8999999999999998e-210

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.5%

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

   6. Taylor expanded in Ec around inf 63.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

   8. Simplified 63.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq -7 \cdot 10^{-181}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 3.9 \cdot 10^{-210}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{Ec}{-KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -2.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev - mu}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -2.1 \cdot 10^{-258}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef - mu}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq 1.3 \cdot 10^{-187}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -2.6e+123)
   (/ NaChar (+ (exp (/ (- Ev mu) KbT)) 1.0))
   (if (<= Ev -2.1e-258)
     (/ NaChar (+ (exp (/ (- Vef mu) KbT)) 1.0))
     (if (<= Ev 1.3e-187)
       (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
       (/ NaChar (+ (exp (/ (- EAccept mu) KbT)) 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -2.6e+123) {
		tmp = NaChar / (exp(((Ev - mu) / KbT)) + 1.0);
	} else if (Ev <= -2.1e-258) {
		tmp = NaChar / (exp(((Vef - mu) / KbT)) + 1.0);
	} else if (Ev <= 1.3e-187) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ev <= (-2.6d+123)) then
        tmp = nachar / (exp(((ev - mu) / kbt)) + 1.0d0)
    else if (ev <= (-2.1d-258)) then
        tmp = nachar / (exp(((vef - mu) / kbt)) + 1.0d0)
    else if (ev <= 1.3d-187) then
        tmp = ndchar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp(((eaccept - mu) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -2.6e+123) {
		tmp = NaChar / (Math.exp(((Ev - mu) / KbT)) + 1.0);
	} else if (Ev <= -2.1e-258) {
		tmp = NaChar / (Math.exp(((Vef - mu) / KbT)) + 1.0);
	} else if (Ev <= 1.3e-187) {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp(((EAccept - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -2.6e+123:
		tmp = NaChar / (math.exp(((Ev - mu) / KbT)) + 1.0)
	elif Ev <= -2.1e-258:
		tmp = NaChar / (math.exp(((Vef - mu) / KbT)) + 1.0)
	elif Ev <= 1.3e-187:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp(((EAccept - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -2.6e+123)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Ev - mu) / KbT)) + 1.0));
	elseif (Ev <= -2.1e-258)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Vef - mu) / KbT)) + 1.0));
	elseif (Ev <= 1.3e-187)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -2.6e+123)
		tmp = NaChar / (exp(((Ev - mu) / KbT)) + 1.0);
	elseif (Ev <= -2.1e-258)
		tmp = NaChar / (exp(((Vef - mu) / KbT)) + 1.0);
	elseif (Ev <= 1.3e-187)
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -2.6e+123], N[(NaChar / N[(N[Exp[N[(N[(Ev - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, -2.1e-258], N[(NaChar / N[(N[Exp[N[(N[(Vef - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, 1.3e-187], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(EAccept - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if Ev < -2.59999999999999985e123

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 56.1%

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

      1. associate-*r/ 56.1%

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

      2. mul-1-neg 56.1%

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

   7. Simplified 56.1%

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

   8. Taylor expanded in NdChar around 0 46.7%

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

   9. Taylor expanded in Ev around inf 43.3%

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

   if -2.59999999999999985e123 < Ev < -2.0999999999999999e-258

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 74.8%

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

      1. associate-*r/ 74.8%

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

      2. mul-1-neg 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in NdChar around 0 57.6%

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

   9. Taylor expanded in Vef around inf 48.5%

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

   if -2.0999999999999999e-258 < Ev < 1.3e-187

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 78.6%

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

   6. Taylor expanded in KbT around inf 44.7%

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

      1. associate-+r+ 55.5%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in Vef around inf 30.4%

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

   10. Taylor expanded in NdChar around inf 41.7%

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

   if 1.3e-187 < Ev

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 69.3%

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

      1. associate-*r/ 69.3%

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

      2. mul-1-neg 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in NdChar around 0 62.4%

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

   9. Taylor expanded in EAccept around inf 45.1%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -2.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev - mu}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -2.1 \cdot 10^{-258}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef - mu}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq 1.3 \cdot 10^{-187}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;NaChar \leq -7.5 \cdot 10^{-50} \lor \neg \left(NaChar \leq 950000\right):\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0)))
   (if (or (<= NaChar -7.5e-50) (not (<= NaChar 950000.0)))
     (/ NaChar t_0)
     (/ NdChar t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((Vef / KbT)) + 1.0;
	double tmp;
	if ((NaChar <= -7.5e-50) || !(NaChar <= 950000.0)) {
		tmp = NaChar / t_0;
	} else {
		tmp = NdChar / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    if ((nachar <= (-7.5d-50)) .or. (.not. (nachar <= 950000.0d0))) then
        tmp = nachar / t_0
    else
        tmp = ndchar / t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp((Vef / KbT)) + 1.0;
	double tmp;
	if ((NaChar <= -7.5e-50) || !(NaChar <= 950000.0)) {
		tmp = NaChar / t_0;
	} else {
		tmp = NdChar / t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	tmp = 0
	if (NaChar <= -7.5e-50) or not (NaChar <= 950000.0):
		tmp = NaChar / t_0
	else:
		tmp = NdChar / t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	tmp = 0.0
	if ((NaChar <= -7.5e-50) || !(NaChar <= 950000.0))
		tmp = Float64(NaChar / t_0);
	else
		tmp = Float64(NdChar / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp((Vef / KbT)) + 1.0;
	tmp = 0.0;
	if ((NaChar <= -7.5e-50) || ~((NaChar <= 950000.0)))
		tmp = NaChar / t_0;
	else
		tmp = NdChar / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[Or[LessEqual[NaChar, -7.5e-50], N[Not[LessEqual[NaChar, 950000.0]], $MachinePrecision]], N[(NaChar / t$95$0), $MachinePrecision], N[(NdChar / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -7.5e-50 or 9.5e5 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 76.2%

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

      1. associate-*r/ 76.2%

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

      2. mul-1-neg 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in NdChar around 0 71.6%

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

   9. Taylor expanded in Vef around inf 55.8%

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

   10. Taylor expanded in mu around 0 46.7%

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

   if -7.5e-50 < NaChar < 9.5e5

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Vef around inf 62.3%

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

   6. Taylor expanded in KbT around inf 47.1%

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

      1. associate-+r+ 68.8%

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

   8. Simplified 47.1%

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

   9. Taylor expanded in Vef around inf 35.7%

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

   10. Taylor expanded in NdChar around inf 51.8%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -7.5 \cdot 10^{-50} \lor \neg \left(NaChar \leq 950000\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 41.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -3 \cdot 10^{+216}:\\ \;\;\;\;\left(NaChar \cdot 0.5 + NdChar \cdot 0.5\right) - -0.25 \cdot \left(NaChar \cdot \frac{mu}{KbT}\right)\\ \mathbf{elif}\;KbT \leq 6.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -3e+216)
   (- (+ (* NaChar 0.5) (* NdChar 0.5)) (* -0.25 (* NaChar (/ mu KbT))))
   (if (<= KbT 6.2e+35)
     (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
     (* 0.5 (+ NdChar NaChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -3e+216) {
		tmp = ((NaChar * 0.5) + (NdChar * 0.5)) - (-0.25 * (NaChar * (mu / KbT)));
	} else if (KbT <= 6.2e+35) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-3d+216)) then
        tmp = ((nachar * 0.5d0) + (ndchar * 0.5d0)) - ((-0.25d0) * (nachar * (mu / kbt)))
    else if (kbt <= 6.2d+35) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = 0.5d0 * (ndchar + nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -3e+216) {
		tmp = ((NaChar * 0.5) + (NdChar * 0.5)) - (-0.25 * (NaChar * (mu / KbT)));
	} else if (KbT <= 6.2e+35) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -3e+216:
		tmp = ((NaChar * 0.5) + (NdChar * 0.5)) - (-0.25 * (NaChar * (mu / KbT)))
	elif KbT <= 6.2e+35:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -3e+216)
		tmp = Float64(Float64(Float64(NaChar * 0.5) + Float64(NdChar * 0.5)) - Float64(-0.25 * Float64(NaChar * Float64(mu / KbT))));
	elseif (KbT <= 6.2e+35)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -3e+216)
		tmp = ((NaChar * 0.5) + (NdChar * 0.5)) - (-0.25 * (NaChar * (mu / KbT)));
	elseif (KbT <= 6.2e+35)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -3e+216], N[(N[(N[(NaChar * 0.5), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision] - N[(-0.25 * N[(NaChar * N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 6.2e+35], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -2.9999999999999998e216

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 95.5%

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

      1. associate-*r/ 95.5%

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

      2. mul-1-neg 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in KbT around inf 57.7%

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

   9. Taylor expanded in mu around inf 68.7%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.25 \cdot \frac{NaChar \cdot mu}{KbT}\right)} + \left(0.5 \cdot NaChar + 0.5 \cdot NdChar\right) \]
   10. Step-by-step derivation

       1. associate-/l* 80.8%

          \[\leadsto -1 \cdot \left(-0.25 \cdot \color{blue}{\left(NaChar \cdot \frac{mu}{KbT}\right)}\right) + \left(0.5 \cdot NaChar + 0.5 \cdot NdChar\right) \]

   11. Simplified 80.8%

       \[\leadsto -1 \cdot \color{blue}{\left(-0.25 \cdot \left(NaChar \cdot \frac{mu}{KbT}\right)\right)} + \left(0.5 \cdot NaChar + 0.5 \cdot NdChar\right) \]

   if -2.9999999999999998e216 < KbT < 6.19999999999999973e35

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 64.5%

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

      1. associate-*r/ 64.5%

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

      2. mul-1-neg 64.5%

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

   7. Simplified 64.5%

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

   8. Taylor expanded in NdChar around 0 62.2%

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

   9. Taylor expanded in Vef around inf 46.6%

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

   10. Taylor expanded in mu around 0 40.2%

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

   if 6.19999999999999973e35 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. mul-1-neg 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in KbT around inf 41.8%

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

      1. distribute-lft-out 41.8%

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

   10. Simplified 41.8%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3 \cdot 10^{+216}:\\ \;\;\;\;\left(NaChar \cdot 0.5 + NdChar \cdot 0.5\right) - -0.25 \cdot \left(NaChar \cdot \frac{mu}{KbT}\right)\\ \mathbf{elif}\;KbT \leq 6.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 30.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -3.2 \cdot 10^{-41} \lor \neg \left(KbT \leq 1550\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 - \left(\frac{mu}{KbT} + \left(-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= KbT -3.2e-41) (not (<= KbT 1550.0)))
   (* 0.5 (+ NdChar NaChar))
   (/
    NaChar
    (-
     1.0
     (+ (/ mu KbT) (- -1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ 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 <= -3.2e-41) || !(KbT <= 1550.0)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NaChar / (1.0 - ((mu / KbT) + (-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (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 <= (-3.2d-41)) .or. (.not. (kbt <= 1550.0d0))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = nachar / (1.0d0 - ((mu / kbt) + ((-1.0d0) - ((eaccept / kbt) + ((vef / kbt) + (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 <= -3.2e-41) || !(KbT <= 1550.0)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NaChar / (1.0 - ((mu / KbT) + (-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (KbT <= -3.2e-41) or not (KbT <= 1550.0):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = NaChar / (1.0 - ((mu / KbT) + (-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((KbT <= -3.2e-41) || !(KbT <= 1550.0))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(NaChar / Float64(1.0 - Float64(Float64(mu / KbT) + Float64(-1.0 - Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + 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 <= -3.2e-41) || ~((KbT <= 1550.0)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = NaChar / (1.0 - ((mu / KbT) + (-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[KbT, -3.2e-41], N[Not[LessEqual[KbT, 1550.0]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(1.0 - N[(N[(mu / KbT), $MachinePrecision] + N[(-1.0 - N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -3.20000000000000012e-41 or 1550 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 72.4%

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

      1. associate-*r/ 72.4%

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

      2. mul-1-neg 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in KbT around inf 39.3%

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

      1. distribute-lft-out 39.3%

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

   10. Simplified 39.3%

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

   if -3.20000000000000012e-41 < KbT < 1550

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 64.0%

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

      1. associate-*r/ 64.0%

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

      2. mul-1-neg 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in NdChar around 0 57.9%

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

   9. Taylor expanded in KbT around inf 19.8%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.2 \cdot 10^{-41} \lor \neg \left(KbT \leq 1550\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 - \left(\frac{mu}{KbT} + \left(-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 30.5% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -1.06 \cdot 10^{-41} \lor \neg \left(KbT \leq 550\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= KbT -1.06e-41) (not (<= KbT 550.0)))
   (* 0.5 (+ NdChar NaChar))
   (/
    NaChar
    (- (+ 2.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT)))) (/ mu KbT)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -1.06e-41) || !(KbT <= 550.0)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((kbt <= (-1.06d-41)) .or. (.not. (kbt <= 550.0d0))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = nachar / ((2.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -1.06e-41) || !(KbT <= 550.0)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (KbT <= -1.06e-41) or not (KbT <= 550.0):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((KbT <= -1.06e-41) || !(KbT <= 550.0))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((KbT <= -1.06e-41) || ~((KbT <= 550.0)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[KbT, -1.06e-41], N[Not[LessEqual[KbT, 550.0]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[(2.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -1.06e-41 or 550 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 72.4%

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

      1. associate-*r/ 72.4%

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

      2. mul-1-neg 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in KbT around inf 39.3%

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

      1. distribute-lft-out 39.3%

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

   10. Simplified 39.3%

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

   if -1.06e-41 < KbT < 550

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 64.0%

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

      1. associate-*r/ 64.0%

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

      2. mul-1-neg 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in NdChar around 0 57.9%

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

   9. Taylor expanded in KbT around inf 19.8%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.06 \cdot 10^{-41} \lor \neg \left(KbT \leq 550\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 27.9% accurate, 45.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(NdChar + NaChar\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in Ec around inf 68.7%

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

   1. associate-*r/ 68.7%

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

   2. mul-1-neg 68.7%

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

7. Simplified 68.7%

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

8. Taylor expanded in KbT around inf 27.6%

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

   1. distribute-lft-out 27.6%

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

10. Simplified 27.6%

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

11. Final simplification 27.6%

    \[\leadsto 0.5 \cdot \left(NdChar + NaChar\right) \]
12. Add Preprocessing

Alternative 26: 18.1% accurate, 76.3× speedup

Math

\[\begin{array}{l} \\ \frac{NaChar}{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in Ec around inf 68.7%

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

   1. associate-*r/ 68.7%

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

   2. mul-1-neg 68.7%

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

7. Simplified 68.7%

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

8. Taylor expanded in NdChar around 0 58.6%

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

9. Taylor expanded in Vef around inf 45.3%

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

10. Taylor expanded in KbT around inf 18.3%

    \[\leadsto \frac{NaChar}{\color{blue}{2}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024083 
(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))))))