Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 35.1s

Alternatives: 21

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 2.3644239522072996e-236
  2. Ec = 4.275103695239931e+209
  3. Vef = 9.77453836619029e-191
  4. EDonor = -1.429128189640755e+167
  5. mu = -9.892291784175356e+182
  6. KbT = 1.2161594923105197e+67
  7. NaChar = 4.7576619670588787e-237
  8. Ev = -3.2081890779470196e+302
  9. EAccept = 1.0448988666038535e-254

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\frac{Vef}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ t_2 := \frac{Vef}{KbT} + \frac{Ev}{KbT}\\ t_3 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{if}\;mu \leq -3.7 \cdot 10^{+146}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;mu \leq -1.1 \cdot 10^{+68}:\\ \;\;\;\;\frac{NdChar}{t\_0} + \frac{NaChar}{t\_0}\\ \mathbf{elif}\;mu \leq -1.8 \cdot 10^{+21}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;mu \leq -4.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.75 \cdot 10^{-210}:\\ \;\;\;\;t\_1 + \frac{NaChar}{mu \cdot \left(\frac{\left(\frac{EAccept}{KbT} + 2\right) + t\_2}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 1.45 \cdot 10^{-147}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{elif}\;mu \leq 5 \cdot 10^{+108}:\\ \;\;\;\;t\_1 + \frac{NaChar}{EAccept \cdot \left(\frac{1}{KbT} + \frac{\left(2 + t\_2\right) - \frac{mu}{KbT}}{EAccept}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (/ Vef KbT))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))
        (t_2 (+ (/ Vef KbT) (/ Ev KbT)))
        (t_3
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ mu (- KbT))))))))
   (if (<= mu -3.7e+146)
     t_3
     (if (<= mu -1.1e+68)
       (+ (/ NdChar t_0) (/ NaChar t_0))
       (if (<= mu -1.8e+21)
         t_3
         (if (<= mu -4.2e-129)
           (-
            (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
            (/ NdChar (- -1.0 (exp (/ EDonor KbT)))))
           (if (<= mu 1.75e-210)
             (+
              t_1
              (/
               NaChar
               (* mu (+ (/ (+ (+ (/ EAccept KbT) 2.0) t_2) mu) (/ -1.0 KbT)))))
             (if (<= mu 1.45e-147)
               (+
                (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))
                (/
                 NdChar
                 (-
                  (+ (+ 2.0 (/ EDonor KbT)) (+ (/ Vef KbT) (/ mu KbT)))
                  (/ Ec KbT))))
               (if (<= mu 5e+108)
                 (+
                  t_1
                  (/
                   NaChar
                   (*
                    EAccept
                    (+ (/ 1.0 KbT) (/ (- (+ 2.0 t_2) (/ mu KbT)) EAccept)))))
                 t_3)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 1.0 + exp((Vef / KbT));
	double t_1 = NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	double t_2 = (Vef / KbT) + (Ev / KbT);
	double t_3 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	double tmp;
	if (mu <= -3.7e+146) {
		tmp = t_3;
	} else if (mu <= -1.1e+68) {
		tmp = (NdChar / t_0) + (NaChar / t_0);
	} else if (mu <= -1.8e+21) {
		tmp = t_3;
	} else if (mu <= -4.2e-129) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	} else if (mu <= 1.75e-210) {
		tmp = t_1 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + t_2) / mu) + (-1.0 / KbT))));
	} else if (mu <= 1.45e-147) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / (((2.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)));
	} else if (mu <= 5e+108) {
		tmp = t_1 + (NaChar / (EAccept * ((1.0 / KbT) + (((2.0 + t_2) - (mu / KbT)) / EAccept))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 + exp((vef / kbt))
    t_1 = ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))
    t_2 = (vef / kbt) + (ev / kbt)
    t_3 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((mu / -kbt))))
    if (mu <= (-3.7d+146)) then
        tmp = t_3
    else if (mu <= (-1.1d+68)) then
        tmp = (ndchar / t_0) + (nachar / t_0)
    else if (mu <= (-1.8d+21)) then
        tmp = t_3
    else if (mu <= (-4.2d-129)) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) - (ndchar / ((-1.0d0) - exp((edonor / kbt))))
    else if (mu <= 1.75d-210) then
        tmp = t_1 + (nachar / (mu * (((((eaccept / kbt) + 2.0d0) + t_2) / mu) + ((-1.0d0) / kbt))))
    else if (mu <= 1.45d-147) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))) + (ndchar / (((2.0d0 + (edonor / kbt)) + ((vef / kbt) + (mu / kbt))) - (ec / kbt)))
    else if (mu <= 5d+108) then
        tmp = t_1 + (nachar / (eaccept * ((1.0d0 / kbt) + (((2.0d0 + t_2) - (mu / kbt)) / eaccept))))
    else
        tmp = t_3
    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 = 1.0 + Math.exp((Vef / KbT));
	double t_1 = NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	double t_2 = (Vef / KbT) + (Ev / KbT);
	double t_3 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	double tmp;
	if (mu <= -3.7e+146) {
		tmp = t_3;
	} else if (mu <= -1.1e+68) {
		tmp = (NdChar / t_0) + (NaChar / t_0);
	} else if (mu <= -1.8e+21) {
		tmp = t_3;
	} else if (mu <= -4.2e-129) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) - (NdChar / (-1.0 - Math.exp((EDonor / KbT))));
	} else if (mu <= 1.75e-210) {
		tmp = t_1 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + t_2) / mu) + (-1.0 / KbT))));
	} else if (mu <= 1.45e-147) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / (((2.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)));
	} else if (mu <= 5e+108) {
		tmp = t_1 + (NaChar / (EAccept * ((1.0 / KbT) + (((2.0 + t_2) - (mu / KbT)) / EAccept))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 1.0 + math.exp((Vef / KbT))
	t_1 = NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))
	t_2 = (Vef / KbT) + (Ev / KbT)
	t_3 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((mu / -KbT))))
	tmp = 0
	if mu <= -3.7e+146:
		tmp = t_3
	elif mu <= -1.1e+68:
		tmp = (NdChar / t_0) + (NaChar / t_0)
	elif mu <= -1.8e+21:
		tmp = t_3
	elif mu <= -4.2e-129:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) - (NdChar / (-1.0 - math.exp((EDonor / KbT))))
	elif mu <= 1.75e-210:
		tmp = t_1 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + t_2) / mu) + (-1.0 / KbT))))
	elif mu <= 1.45e-147:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / (((2.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)))
	elif mu <= 5e+108:
		tmp = t_1 + (NaChar / (EAccept * ((1.0 / KbT) + (((2.0 + t_2) - (mu / KbT)) / EAccept))))
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(1.0 + exp(Float64(Vef / KbT)))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT))))
	t_2 = Float64(Float64(Vef / KbT) + Float64(Ev / KbT))
	t_3 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))))
	tmp = 0.0
	if (mu <= -3.7e+146)
		tmp = t_3;
	elseif (mu <= -1.1e+68)
		tmp = Float64(Float64(NdChar / t_0) + Float64(NaChar / t_0));
	elseif (mu <= -1.8e+21)
		tmp = t_3;
	elseif (mu <= -4.2e-129)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(EDonor / KbT)))));
	elseif (mu <= 1.75e-210)
		tmp = Float64(t_1 + Float64(NaChar / Float64(mu * Float64(Float64(Float64(Float64(Float64(EAccept / KbT) + 2.0) + t_2) / mu) + Float64(-1.0 / KbT)))));
	elseif (mu <= 1.45e-147)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT)))) + Float64(NdChar / Float64(Float64(Float64(2.0 + Float64(EDonor / KbT)) + Float64(Float64(Vef / KbT) + Float64(mu / KbT))) - Float64(Ec / KbT))));
	elseif (mu <= 5e+108)
		tmp = Float64(t_1 + Float64(NaChar / Float64(EAccept * Float64(Float64(1.0 / KbT) + Float64(Float64(Float64(2.0 + t_2) - Float64(mu / KbT)) / EAccept)))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 1.0 + exp((Vef / KbT));
	t_1 = NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	t_2 = (Vef / KbT) + (Ev / KbT);
	t_3 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	tmp = 0.0;
	if (mu <= -3.7e+146)
		tmp = t_3;
	elseif (mu <= -1.1e+68)
		tmp = (NdChar / t_0) + (NaChar / t_0);
	elseif (mu <= -1.8e+21)
		tmp = t_3;
	elseif (mu <= -4.2e-129)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	elseif (mu <= 1.75e-210)
		tmp = t_1 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + t_2) / mu) + (-1.0 / KbT))));
	elseif (mu <= 1.45e-147)
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / (((2.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)));
	elseif (mu <= 5e+108)
		tmp = t_1 + (NaChar / (EAccept * ((1.0 / KbT) + (((2.0 + t_2) - (mu / KbT)) / EAccept))));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -3.7e+146], t$95$3, If[LessEqual[mu, -1.1e+68], N[(N[(NdChar / t$95$0), $MachinePrecision] + N[(NaChar / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, -1.8e+21], t$95$3, If[LessEqual[mu, -4.2e-129], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.75e-210], N[(t$95$1 + N[(NaChar / N[(mu * N[(N[(N[(N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / mu), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.45e-147], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[(N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 5e+108], N[(t$95$1 + N[(NaChar / N[(EAccept * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(N[(2.0 + t$95$2), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision] / EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if mu < -3.70000000000000004e146 or -1.09999999999999994e68 < mu < -1.8e21 or 4.99999999999999991e108 < 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 88.4%

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

   6. Taylor expanded in mu around inf 80.6%

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

      1. associate-*r/ 37.7%

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

      2. mul-1-neg 37.7%

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

   8. Simplified 80.6%

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

   if -3.70000000000000004e146 < mu < -1.09999999999999994e68

   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 88.7%

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

   6. Taylor expanded in EAccept around 0 82.7%

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

   7. Taylor expanded in Vef around inf 83.7%

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

   if -1.8e21 < mu < -4.2e-129

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 80.1%

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

   6. Taylor expanded in EDonor around inf 75.9%

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

   if -4.2e-129 < mu < 1.75000000000000008e-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 72.1%

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

      1. associate-+r+ 72.1%

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

      2. +-commutative 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in mu around -inf 84.9%

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

      1. associate-*r* 84.9%

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

      2. mul-1-neg 84.9%

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

      3. +-commutative 84.9%

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

      4. mul-1-neg 84.9%

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

      5. unsub-neg 84.9%

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

      6. associate-+r+ 84.9%

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

   10. Simplified 84.9%

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

   if 1.75000000000000008e-210 < mu < 1.4500000000000001e-147

   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 94.0%

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

      1. associate-+r+ 94.0%

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

   7. Simplified 94.0%

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

   if 1.4500000000000001e-147 < mu < 4.99999999999999991e108

   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 69.2%

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

      1. associate-+r+ 69.2%

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

      2. +-commutative 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in EAccept around -inf 75.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{-1 \cdot \left(EAccept \cdot \left(-1 \cdot \frac{\left(2 + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}}{EAccept} - \frac{1}{KbT}\right)\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -3.7 \cdot 10^{+146}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq -1.1 \cdot 10^{+68}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq -1.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq -4.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.75 \cdot 10^{-210}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{mu \cdot \left(\frac{\left(\frac{EAccept}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 1.45 \cdot 10^{-147}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{elif}\;mu \leq 5 \cdot 10^{+108}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{EAccept \cdot \left(\frac{1}{KbT} + \frac{\left(2 + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}{EAccept}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{if}\;mu \leq -7.5 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;mu \leq -1.4 \cdot 10^{-213}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;mu \leq 3 \cdot 10^{-253}:\\ \;\;\;\;t\_0 + \frac{NaChar}{mu \cdot \left(\frac{\left(\frac{EAccept}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 5.5 \cdot 10^{+89}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;mu \leq 1.18 \cdot 10^{+150}:\\ \;\;\;\;t\_0 + \frac{NaChar}{EAccept \cdot \left(\frac{2}{EAccept} - \left(\frac{\frac{mu}{EAccept}}{KbT} - \left(\frac{Vef}{KbT \cdot EAccept} - \left(\frac{-1}{KbT} - \frac{\frac{Ev}{EAccept}}{KbT}\right)\right)\right)\right)}\\ \mathbf{elif}\;mu \leq 3.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Ec}{-KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ mu (- KbT)))))))
        (t_2
         (-
          (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ Vef Ev) mu) KbT)))))))
   (if (<= mu -7.5e+166)
     t_1
     (if (<= mu -1.4e-213)
       t_2
       (if (<= mu 3e-253)
         (+
          t_0
          (/
           NaChar
           (*
            mu
            (+
             (/ (+ (+ (/ EAccept KbT) 2.0) (+ (/ Vef KbT) (/ Ev KbT))) mu)
             (/ -1.0 KbT)))))
         (if (<= mu 5.5e+89)
           t_2
           (if (<= mu 1.18e+150)
             (+
              t_0
              (/
               NaChar
               (*
                EAccept
                (-
                 (/ 2.0 EAccept)
                 (-
                  (/ (/ mu EAccept) KbT)
                  (-
                   (/ Vef (* KbT EAccept))
                   (- (/ -1.0 KbT) (/ (/ Ev EAccept) KbT))))))))
             (if (<= mu 3.8e+174)
               (+
                (/ NdChar (+ 1.0 (exp (/ Ec (- KbT)))))
                (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
               t_1))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	double t_2 = (NdChar / (1.0 + exp((Vef / KbT)))) - (NaChar / (-1.0 - exp((((Vef + Ev) - mu) / KbT))));
	double tmp;
	if (mu <= -7.5e+166) {
		tmp = t_1;
	} else if (mu <= -1.4e-213) {
		tmp = t_2;
	} else if (mu <= 3e-253) {
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))));
	} else if (mu <= 5.5e+89) {
		tmp = t_2;
	} else if (mu <= 1.18e+150) {
		tmp = t_0 + (NaChar / (EAccept * ((2.0 / EAccept) - (((mu / EAccept) / KbT) - ((Vef / (KbT * EAccept)) - ((-1.0 / KbT) - ((Ev / EAccept) / KbT)))))));
	} else if (mu <= 3.8e+174) {
		tmp = (NdChar / (1.0 + exp((Ec / -KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((mu / -kbt))))
    t_2 = (ndchar / (1.0d0 + exp((vef / kbt)))) - (nachar / ((-1.0d0) - exp((((vef + ev) - mu) / kbt))))
    if (mu <= (-7.5d+166)) then
        tmp = t_1
    else if (mu <= (-1.4d-213)) then
        tmp = t_2
    else if (mu <= 3d-253) then
        tmp = t_0 + (nachar / (mu * (((((eaccept / kbt) + 2.0d0) + ((vef / kbt) + (ev / kbt))) / mu) + ((-1.0d0) / kbt))))
    else if (mu <= 5.5d+89) then
        tmp = t_2
    else if (mu <= 1.18d+150) then
        tmp = t_0 + (nachar / (eaccept * ((2.0d0 / eaccept) - (((mu / eaccept) / kbt) - ((vef / (kbt * eaccept)) - (((-1.0d0) / kbt) - ((ev / eaccept) / kbt)))))))
    else if (mu <= 3.8d+174) then
        tmp = (ndchar / (1.0d0 + exp((ec / -kbt)))) + (nachar / (1.0d0 + exp((ev / kbt))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	double t_2 = (NdChar / (1.0 + Math.exp((Vef / KbT)))) - (NaChar / (-1.0 - Math.exp((((Vef + Ev) - mu) / KbT))));
	double tmp;
	if (mu <= -7.5e+166) {
		tmp = t_1;
	} else if (mu <= -1.4e-213) {
		tmp = t_2;
	} else if (mu <= 3e-253) {
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))));
	} else if (mu <= 5.5e+89) {
		tmp = t_2;
	} else if (mu <= 1.18e+150) {
		tmp = t_0 + (NaChar / (EAccept * ((2.0 / EAccept) - (((mu / EAccept) / KbT) - ((Vef / (KbT * EAccept)) - ((-1.0 / KbT) - ((Ev / EAccept) / KbT)))))));
	} else if (mu <= 3.8e+174) {
		tmp = (NdChar / (1.0 + Math.exp((Ec / -KbT)))) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((mu / -KbT))))
	t_2 = (NdChar / (1.0 + math.exp((Vef / KbT)))) - (NaChar / (-1.0 - math.exp((((Vef + Ev) - mu) / KbT))))
	tmp = 0
	if mu <= -7.5e+166:
		tmp = t_1
	elif mu <= -1.4e-213:
		tmp = t_2
	elif mu <= 3e-253:
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))))
	elif mu <= 5.5e+89:
		tmp = t_2
	elif mu <= 1.18e+150:
		tmp = t_0 + (NaChar / (EAccept * ((2.0 / EAccept) - (((mu / EAccept) / KbT) - ((Vef / (KbT * EAccept)) - ((-1.0 / KbT) - ((Ev / EAccept) / KbT)))))))
	elif mu <= 3.8e+174:
		tmp = (NdChar / (1.0 + math.exp((Ec / -KbT)))) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))))
	tmp = 0.0
	if (mu <= -7.5e+166)
		tmp = t_1;
	elseif (mu <= -1.4e-213)
		tmp = t_2;
	elseif (mu <= 3e-253)
		tmp = Float64(t_0 + Float64(NaChar / Float64(mu * Float64(Float64(Float64(Float64(Float64(EAccept / KbT) + 2.0) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) / mu) + Float64(-1.0 / KbT)))));
	elseif (mu <= 5.5e+89)
		tmp = t_2;
	elseif (mu <= 1.18e+150)
		tmp = Float64(t_0 + Float64(NaChar / Float64(EAccept * Float64(Float64(2.0 / EAccept) - Float64(Float64(Float64(mu / EAccept) / KbT) - Float64(Float64(Vef / Float64(KbT * EAccept)) - Float64(Float64(-1.0 / KbT) - Float64(Float64(Ev / EAccept) / KbT))))))));
	elseif (mu <= 3.8e+174)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Ec / Float64(-KbT))))) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	t_1 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	t_2 = (NdChar / (1.0 + exp((Vef / KbT)))) - (NaChar / (-1.0 - exp((((Vef + Ev) - mu) / KbT))));
	tmp = 0.0;
	if (mu <= -7.5e+166)
		tmp = t_1;
	elseif (mu <= -1.4e-213)
		tmp = t_2;
	elseif (mu <= 3e-253)
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))));
	elseif (mu <= 5.5e+89)
		tmp = t_2;
	elseif (mu <= 1.18e+150)
		tmp = t_0 + (NaChar / (EAccept * ((2.0 / EAccept) - (((mu / EAccept) / KbT) - ((Vef / (KbT * EAccept)) - ((-1.0 / KbT) - ((Ev / EAccept) / KbT)))))));
	elseif (mu <= 3.8e+174)
		tmp = (NdChar / (1.0 + exp((Ec / -KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -7.5e+166], t$95$1, If[LessEqual[mu, -1.4e-213], t$95$2, If[LessEqual[mu, 3e-253], N[(t$95$0 + N[(NaChar / N[(mu * N[(N[(N[(N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 5.5e+89], t$95$2, If[LessEqual[mu, 1.18e+150], N[(t$95$0 + N[(NaChar / N[(EAccept * N[(N[(2.0 / EAccept), $MachinePrecision] - N[(N[(N[(mu / EAccept), $MachinePrecision] / KbT), $MachinePrecision] - N[(N[(Vef / N[(KbT * EAccept), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 / KbT), $MachinePrecision] - N[(N[(Ev / EAccept), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 3.8e+174], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if mu < -7.50000000000000029e166 or 3.8000000000000002e174 < 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 90.5%

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

   6. Taylor expanded in mu around inf 82.1%

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

      1. associate-*r/ 35.0%

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

      2. mul-1-neg 35.0%

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

   8. Simplified 82.1%

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

   if -7.50000000000000029e166 < mu < -1.4e-213 or 3.0000000000000002e-253 < mu < 5.49999999999999976e89

   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 83.2%

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

   6. Taylor expanded in EAccept around 0 77.3%

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

   if -1.4e-213 < mu < 3.0000000000000002e-253

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 77.4%

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

      1. associate-+r+ 77.4%

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

      2. +-commutative 77.4%

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

   7. Simplified 77.4%

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

   8. Taylor expanded in mu around -inf 86.8%

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

      1. associate-*r* 86.8%

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

      2. mul-1-neg 86.8%

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

      3. +-commutative 86.8%

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

      4. mul-1-neg 86.8%

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

      5. unsub-neg 86.8%

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

      6. associate-+r+ 86.8%

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

   10. Simplified 86.8%

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

   if 5.49999999999999976e89 < mu < 1.18e150

   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 91.8%

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

      1. associate-+r+ 91.8%

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

      2. +-commutative 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in EAccept around inf 75.2%

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

      1. associate--l+ 75.2%

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

      2. associate-*r/ 75.2%

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

      3. metadata-eval 75.2%

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

      4. associate-+r+ 75.2%

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

      5. associate-/r* 91.8%

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

      6. *-commutative 91.8%

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

      7. associate-/r* 91.8%

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

   10. Simplified 91.8%

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

   if 1.18e150 < mu < 3.8000000000000002e174

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 74.5%

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

   6. Taylor expanded in Ec around inf 74.5%

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

      1. associate-*r/ 74.5%

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

      2. mul-1-neg 74.5%

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

   8. Simplified 74.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -7.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq -1.4 \cdot 10^{-213}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 3 \cdot 10^{-253}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{mu \cdot \left(\frac{\left(\frac{EAccept}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 5.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.18 \cdot 10^{+150}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{EAccept \cdot \left(\frac{2}{EAccept} - \left(\frac{\frac{mu}{EAccept}}{KbT} - \left(\frac{Vef}{KbT \cdot EAccept} - \left(\frac{-1}{KbT} - \frac{\frac{Ev}{EAccept}}{KbT}\right)\right)\right)\right)}\\ \mathbf{elif}\;mu \leq 3.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Ec}{-KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\\ \mathbf{if}\;Vef \leq -4.65 \cdot 10^{+33}:\\ \;\;\;\;t\_1 + t\_0\\ \mathbf{elif}\;Vef \leq 1.1 \cdot 10^{-249}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.45 \cdot 10^{+64}:\\ \;\;\;\;t\_1 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{NaChar}{-1 - e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))))
   (if (<= Vef -4.65e+33)
     (+ t_1 t_0)
     (if (<= Vef 1.1e-249)
       (+
        (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
        (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
       (if (<= Vef 1.45e+64)
         (+ t_1 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
         (- t_0 (/ NaChar (- -1.0 (exp (/ (- (+ Vef Ev) mu) KbT))))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((Vef / KbT)));
	double t_1 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (Vef <= -4.65e+33) {
		tmp = t_1 + t_0;
	} else if (Vef <= 1.1e-249) {
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (Vef <= 1.45e+64) {
		tmp = t_1 + (NdChar / (1.0 + exp((mu / KbT))));
	} else {
		tmp = t_0 - (NaChar / (-1.0 - exp((((Vef + Ev) - mu) / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((vef / kbt)))
    t_1 = nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))
    if (vef <= (-4.65d+33)) then
        tmp = t_1 + t_0
    else if (vef <= 1.1d-249) then
        tmp = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else if (vef <= 1.45d+64) then
        tmp = t_1 + (ndchar / (1.0d0 + exp((mu / kbt))))
    else
        tmp = t_0 - (nachar / ((-1.0d0) - exp((((vef + ev) - mu) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((Vef / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (Vef <= -4.65e+33) {
		tmp = t_1 + t_0;
	} else if (Vef <= 1.1e-249) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else if (Vef <= 1.45e+64) {
		tmp = t_1 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else {
		tmp = t_0 - (NaChar / (-1.0 - Math.exp((((Vef + Ev) - mu) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((Vef / KbT)))
	t_1 = NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))
	tmp = 0
	if Vef <= -4.65e+33:
		tmp = t_1 + t_0
	elif Vef <= 1.1e-249:
		tmp = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	elif Vef <= 1.45e+64:
		tmp = t_1 + (NdChar / (1.0 + math.exp((mu / KbT))))
	else:
		tmp = t_0 - (NaChar / (-1.0 - math.exp((((Vef + Ev) - mu) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT))))
	tmp = 0.0
	if (Vef <= -4.65e+33)
		tmp = Float64(t_1 + t_0);
	elseif (Vef <= 1.1e-249)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (Vef <= 1.45e+64)
		tmp = Float64(t_1 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	else
		tmp = Float64(t_0 - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((Vef / KbT)));
	t_1 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	tmp = 0.0;
	if (Vef <= -4.65e+33)
		tmp = t_1 + t_0;
	elseif (Vef <= 1.1e-249)
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	elseif (Vef <= 1.45e+64)
		tmp = t_1 + (NdChar / (1.0 + exp((mu / KbT))));
	else
		tmp = t_0 - (NaChar / (-1.0 - exp((((Vef + Ev) - mu) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -4.65e+33], N[(t$95$1 + t$95$0), $MachinePrecision], If[LessEqual[Vef, 1.1e-249], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 1.45e+64], N[(t$95$1 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -4.65000000000000001e33

   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 83.4%

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

   if -4.65000000000000001e33 < Vef < 1.1e-249

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 80.6%

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

   if 1.1e-249 < Vef < 1.44999999999999997e64

   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 90.1%

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

   if 1.44999999999999997e64 < 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 85.4%

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

   6. Taylor expanded in EAccept around 0 80.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -4.65 \cdot 10^{+33}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.1 \cdot 10^{-249}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.45 \cdot 10^{+64}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -6.5 \cdot 10^{+196}:\\ \;\;\;\;t\_1 + KbT \cdot \frac{NdChar}{EDonor - \left(Ec - \left(mu + Vef\right)\right)}\\ \mathbf{elif}\;NaChar \leq -2.4 \cdot 10^{+48}:\\ \;\;\;\;t\_0 + \frac{NaChar}{mu \cdot \left(\frac{\left(\frac{EAccept}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -6.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;t\_0 - \frac{NaChar}{\frac{mu - \left(\left(Ev + EAccept\right) + \left(Vef + KbT \cdot 2\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{-1}{\frac{\frac{Ec - \left(EDonor + \left(mu + Vef\right)\right)}{KbT} - 2}{NdChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))))
   (if (<= NaChar -6.5e+196)
     (+ t_1 (* KbT (/ NdChar (- EDonor (- Ec (+ mu Vef))))))
     (if (<= NaChar -2.4e+48)
       (+
        t_0
        (/
         NaChar
         (*
          mu
          (+
           (/ (+ (+ (/ EAccept KbT) 2.0) (+ (/ Vef KbT) (/ Ev KbT))) mu)
           (/ -1.0 KbT)))))
       (if (<= NaChar -6.6e-21)
         (+
          (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NaChar (+ 1.0 (exp (/ mu (- KbT))))))
         (if (<= NaChar 1.9e-26)
           (-
            t_0
            (/ NaChar (/ (- mu (+ (+ Ev EAccept) (+ Vef (* KbT 2.0)))) KbT)))
           (+
            t_1
            (/
             -1.0
             (/ (- (/ (- Ec (+ EDonor (+ mu Vef))) KbT) 2.0) NdChar)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	double t_1 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (NaChar <= -6.5e+196) {
		tmp = t_1 + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))));
	} else if (NaChar <= -2.4e+48) {
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))));
	} else if (NaChar <= -6.6e-21) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	} else if (NaChar <= 1.9e-26) {
		tmp = t_0 - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT));
	} else {
		tmp = t_1 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))
    t_1 = nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))
    if (nachar <= (-6.5d+196)) then
        tmp = t_1 + (kbt * (ndchar / (edonor - (ec - (mu + vef)))))
    else if (nachar <= (-2.4d+48)) then
        tmp = t_0 + (nachar / (mu * (((((eaccept / kbt) + 2.0d0) + ((vef / kbt) + (ev / kbt))) / mu) + ((-1.0d0) / kbt))))
    else if (nachar <= (-6.6d-21)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / (1.0d0 + exp((mu / -kbt))))
    else if (nachar <= 1.9d-26) then
        tmp = t_0 - (nachar / ((mu - ((ev + eaccept) + (vef + (kbt * 2.0d0)))) / kbt))
    else
        tmp = t_1 + ((-1.0d0) / ((((ec - (edonor + (mu + vef))) / kbt) - 2.0d0) / ndchar))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (NaChar <= -6.5e+196) {
		tmp = t_1 + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))));
	} else if (NaChar <= -2.4e+48) {
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))));
	} else if (NaChar <= -6.6e-21) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	} else if (NaChar <= 1.9e-26) {
		tmp = t_0 - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT));
	} else {
		tmp = t_1 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))
	t_1 = NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))
	tmp = 0
	if NaChar <= -6.5e+196:
		tmp = t_1 + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))))
	elif NaChar <= -2.4e+48:
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))))
	elif NaChar <= -6.6e-21:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / (1.0 + math.exp((mu / -KbT))))
	elif NaChar <= 1.9e-26:
		tmp = t_0 - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT))
	else:
		tmp = t_1 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT))))
	tmp = 0.0
	if (NaChar <= -6.5e+196)
		tmp = Float64(t_1 + Float64(KbT * Float64(NdChar / Float64(EDonor - Float64(Ec - Float64(mu + Vef))))));
	elseif (NaChar <= -2.4e+48)
		tmp = Float64(t_0 + Float64(NaChar / Float64(mu * Float64(Float64(Float64(Float64(Float64(EAccept / KbT) + 2.0) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) / mu) + Float64(-1.0 / KbT)))));
	elseif (NaChar <= -6.6e-21)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))));
	elseif (NaChar <= 1.9e-26)
		tmp = Float64(t_0 - Float64(NaChar / Float64(Float64(mu - Float64(Float64(Ev + EAccept) + Float64(Vef + Float64(KbT * 2.0)))) / KbT)));
	else
		tmp = Float64(t_1 + Float64(-1.0 / Float64(Float64(Float64(Float64(Ec - Float64(EDonor + Float64(mu + Vef))) / KbT) - 2.0) / NdChar)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	t_1 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	tmp = 0.0;
	if (NaChar <= -6.5e+196)
		tmp = t_1 + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))));
	elseif (NaChar <= -2.4e+48)
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))));
	elseif (NaChar <= -6.6e-21)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	elseif (NaChar <= 1.9e-26)
		tmp = t_0 - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT));
	else
		tmp = t_1 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -6.5e+196], N[(t$95$1 + N[(KbT * N[(NdChar / N[(EDonor - N[(Ec - N[(mu + Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, -2.4e+48], N[(t$95$0 + N[(NaChar / N[(mu * N[(N[(N[(N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, -6.6e-21], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.9e-26], N[(t$95$0 - N[(NaChar / N[(N[(mu - N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef + N[(KbT * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.0 / N[(N[(N[(N[(Ec - N[(EDonor + N[(mu + Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision] - 2.0), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NaChar < -6.49999999999999968e196

   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 54.9%

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

      1. associate-+r+ 54.9%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in KbT around 0 55.0%

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

      1. associate-/l* 61.3%

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

      2. associate--l+ 61.3%

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

      3. +-commutative 61.3%

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

   10. Simplified 61.3%

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

   if -6.49999999999999968e196 < NaChar < -2.4000000000000001e48

   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 59.0%

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

      1. associate-+r+ 59.0%

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

      2. +-commutative 59.0%

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

   7. Simplified 59.0%

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

   8. Taylor expanded in mu around -inf 70.4%

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

      1. associate-*r* 70.4%

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

      2. mul-1-neg 70.4%

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

      3. +-commutative 70.4%

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

      4. mul-1-neg 70.4%

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

      5. unsub-neg 70.4%

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

      6. associate-+r+ 70.4%

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

   10. Simplified 70.4%

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

   if -2.4000000000000001e48 < NaChar < -6.60000000000000018e-21

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 84.9%

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

   6. Taylor expanded in mu around inf 78.2%

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

      1. associate-*r/ 47.5%

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

      2. mul-1-neg 47.5%

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

   8. Simplified 78.2%

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

   if -6.60000000000000018e-21 < NaChar < 1.90000000000000007e-26

   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 74.1%

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

      1. associate-+r+ 74.1%

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

      2. +-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in KbT around 0 80.1%

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

      1. associate-+r+ 80.1%

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

   10. Simplified 80.1%

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

   if 1.90000000000000007e-26 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 70.7%

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

      1. associate-+r+ 70.7%

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

   7. Simplified 70.7%

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

      1. clear-num 71.4%

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

      2. inv-pow 71.4%

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

      3. associate-+r+ 71.4%

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

   9. Applied egg-rr 71.4%

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

       1. unpow-1 71.4%

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

       2. associate-+l+ 71.4%

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

       3. associate-+r+ 71.4%

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

       4. associate--l+ 71.4%

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

       5. +-commutative 71.4%

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

   11. Simplified 71.4%

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

   12. Taylor expanded in KbT around 0 76.8%

       \[\leadsto \frac{1}{\frac{2 + \color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}{NdChar}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -6.5 \cdot 10^{+196}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + KbT \cdot \frac{NdChar}{EDonor - \left(Ec - \left(mu + Vef\right)\right)}\\ \mathbf{elif}\;NaChar \leq -2.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{mu \cdot \left(\frac{\left(\frac{EAccept}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -6.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} - \frac{NaChar}{\frac{mu - \left(\left(Ev + EAccept\right) + \left(Vef + KbT \cdot 2\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{-1}{\frac{\frac{Ec - \left(EDonor + \left(mu + Vef\right)\right)}{KbT} - 2}{NdChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2 \cdot 10^{+22} \lor \neg \left(NdChar \leq 2.5 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept \cdot \left(1 - \frac{mu - \left(Vef + Ev\right)}{EAccept}\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -2e+22) (not (<= NdChar 2.5e+171)))
   (+
    (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
    (/ NaChar (/ (* EAccept (- 1.0 (/ (- mu (+ Vef Ev)) EAccept))) KbT)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))
    (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2e+22) || !(NdChar <= 2.5e+171)) {
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / ((EAccept * (1.0 - ((mu - (Vef + Ev)) / EAccept))) / KbT));
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / (1.0 + exp((Vef / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-2d+22)) .or. (.not. (ndchar <= 2.5d+171))) then
        tmp = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) + (nachar / ((eaccept * (1.0d0 - ((mu - (vef + ev)) / eaccept))) / kbt))
    else
        tmp = (nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))) + (ndchar / (1.0d0 + exp((vef / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2e+22) || !(NdChar <= 2.5e+171)) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / ((EAccept * (1.0 - ((mu - (Vef + Ev)) / EAccept))) / KbT));
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -2e+22) or not (NdChar <= 2.5e+171):
		tmp = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / ((EAccept * (1.0 - ((mu - (Vef + Ev)) / EAccept))) / KbT))
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / (1.0 + math.exp((Vef / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -2e+22) || !(NdChar <= 2.5e+171))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) + Float64(NaChar / Float64(Float64(EAccept * Float64(1.0 - Float64(Float64(mu - Float64(Vef + Ev)) / EAccept))) / KbT)));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -2e+22) || ~((NdChar <= 2.5e+171)))
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / ((EAccept * (1.0 - ((mu - (Vef + Ev)) / EAccept))) / KbT));
	else
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / (1.0 + exp((Vef / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -2e+22], N[Not[LessEqual[NdChar, 2.5e+171]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(EAccept * N[(1.0 - N[(N[(mu - N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] / EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2e22 or 2.5000000000000002e171 < 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 73.2%

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

      1. associate-+r+ 73.2%

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

      2. +-commutative 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in EAccept around -inf 80.9%

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

   9. Taylor expanded in KbT around 0 85.0%

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

   if -2e22 < NdChar < 2.5000000000000002e171

   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 78.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2 \cdot 10^{+22} \lor \neg \left(NdChar \leq 2.5 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept \cdot \left(1 - \frac{mu - \left(Vef + Ev\right)}{EAccept}\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\\ \mathbf{if}\;Vef \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;t\_1 + t\_0\\ \mathbf{elif}\;Vef \leq 1.3 \cdot 10^{+64}:\\ \;\;\;\;t\_1 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{NaChar}{-1 - e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))))
   (if (<= Vef -2.5e+37)
     (+ t_1 t_0)
     (if (<= Vef 1.3e+64)
       (+ t_1 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
       (- t_0 (/ NaChar (- -1.0 (exp (/ (- (+ Vef Ev) mu) KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((Vef / KbT)));
	double t_1 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (Vef <= -2.5e+37) {
		tmp = t_1 + t_0;
	} else if (Vef <= 1.3e+64) {
		tmp = t_1 + (NdChar / (1.0 + exp((mu / KbT))));
	} else {
		tmp = t_0 - (NaChar / (-1.0 - exp((((Vef + Ev) - mu) / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((vef / kbt)))
    t_1 = nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))
    if (vef <= (-2.5d+37)) then
        tmp = t_1 + t_0
    else if (vef <= 1.3d+64) then
        tmp = t_1 + (ndchar / (1.0d0 + exp((mu / kbt))))
    else
        tmp = t_0 - (nachar / ((-1.0d0) - exp((((vef + ev) - mu) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((Vef / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (Vef <= -2.5e+37) {
		tmp = t_1 + t_0;
	} else if (Vef <= 1.3e+64) {
		tmp = t_1 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else {
		tmp = t_0 - (NaChar / (-1.0 - Math.exp((((Vef + Ev) - mu) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((Vef / KbT)))
	t_1 = NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))
	tmp = 0
	if Vef <= -2.5e+37:
		tmp = t_1 + t_0
	elif Vef <= 1.3e+64:
		tmp = t_1 + (NdChar / (1.0 + math.exp((mu / KbT))))
	else:
		tmp = t_0 - (NaChar / (-1.0 - math.exp((((Vef + Ev) - mu) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT))))
	tmp = 0.0
	if (Vef <= -2.5e+37)
		tmp = Float64(t_1 + t_0);
	elseif (Vef <= 1.3e+64)
		tmp = Float64(t_1 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	else
		tmp = Float64(t_0 - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((Vef / KbT)));
	t_1 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	tmp = 0.0;
	if (Vef <= -2.5e+37)
		tmp = t_1 + t_0;
	elseif (Vef <= 1.3e+64)
		tmp = t_1 + (NdChar / (1.0 + exp((mu / KbT))));
	else
		tmp = t_0 - (NaChar / (-1.0 - exp((((Vef + Ev) - mu) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -2.5e+37], N[(t$95$1 + t$95$0), $MachinePrecision], If[LessEqual[Vef, 1.3e+64], N[(t$95$1 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -2.49999999999999994e37

   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 85.0%

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

   if -2.49999999999999994e37 < Vef < 1.29999999999999998e64

   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 82.5%

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

   if 1.29999999999999998e64 < 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 85.4%

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

   6. Taylor expanded in EAccept around 0 80.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.3 \cdot 10^{+64}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -3.8 \cdot 10^{+196}:\\ \;\;\;\;t\_1 + KbT \cdot \frac{NdChar}{EDonor - \left(Ec - \left(mu + Vef\right)\right)}\\ \mathbf{elif}\;NaChar \leq -3.5 \cdot 10^{+57}:\\ \;\;\;\;t\_0 + \frac{NaChar}{mu \cdot \left(\frac{\left(\frac{EAccept}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -1.55 \cdot 10^{-46} \lor \neg \left(NaChar \leq 1.6 \cdot 10^{-26}\right):\\ \;\;\;\;t\_1 + \frac{-1}{\frac{\frac{Ec - \left(EDonor + \left(mu + Vef\right)\right)}{KbT} - 2}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{NaChar}{\frac{mu - \left(\left(Ev + EAccept\right) + \left(Vef + KbT \cdot 2\right)\right)}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))))
   (if (<= NaChar -3.8e+196)
     (+ t_1 (* KbT (/ NdChar (- EDonor (- Ec (+ mu Vef))))))
     (if (<= NaChar -3.5e+57)
       (+
        t_0
        (/
         NaChar
         (*
          mu
          (+
           (/ (+ (+ (/ EAccept KbT) 2.0) (+ (/ Vef KbT) (/ Ev KbT))) mu)
           (/ -1.0 KbT)))))
       (if (or (<= NaChar -1.55e-46) (not (<= NaChar 1.6e-26)))
         (+
          t_1
          (/ -1.0 (/ (- (/ (- Ec (+ EDonor (+ mu Vef))) KbT) 2.0) NdChar)))
         (-
          t_0
          (/
           NaChar
           (/ (- mu (+ (+ Ev EAccept) (+ Vef (* KbT 2.0)))) KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	double t_1 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (NaChar <= -3.8e+196) {
		tmp = t_1 + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))));
	} else if (NaChar <= -3.5e+57) {
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))));
	} else if ((NaChar <= -1.55e-46) || !(NaChar <= 1.6e-26)) {
		tmp = t_1 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	} else {
		tmp = t_0 - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))
    t_1 = nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))
    if (nachar <= (-3.8d+196)) then
        tmp = t_1 + (kbt * (ndchar / (edonor - (ec - (mu + vef)))))
    else if (nachar <= (-3.5d+57)) then
        tmp = t_0 + (nachar / (mu * (((((eaccept / kbt) + 2.0d0) + ((vef / kbt) + (ev / kbt))) / mu) + ((-1.0d0) / kbt))))
    else if ((nachar <= (-1.55d-46)) .or. (.not. (nachar <= 1.6d-26))) then
        tmp = t_1 + ((-1.0d0) / ((((ec - (edonor + (mu + vef))) / kbt) - 2.0d0) / ndchar))
    else
        tmp = t_0 - (nachar / ((mu - ((ev + eaccept) + (vef + (kbt * 2.0d0)))) / kbt))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (NaChar <= -3.8e+196) {
		tmp = t_1 + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))));
	} else if (NaChar <= -3.5e+57) {
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))));
	} else if ((NaChar <= -1.55e-46) || !(NaChar <= 1.6e-26)) {
		tmp = t_1 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	} else {
		tmp = t_0 - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))
	t_1 = NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))
	tmp = 0
	if NaChar <= -3.8e+196:
		tmp = t_1 + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))))
	elif NaChar <= -3.5e+57:
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))))
	elif (NaChar <= -1.55e-46) or not (NaChar <= 1.6e-26):
		tmp = t_1 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar))
	else:
		tmp = t_0 - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT))))
	tmp = 0.0
	if (NaChar <= -3.8e+196)
		tmp = Float64(t_1 + Float64(KbT * Float64(NdChar / Float64(EDonor - Float64(Ec - Float64(mu + Vef))))));
	elseif (NaChar <= -3.5e+57)
		tmp = Float64(t_0 + Float64(NaChar / Float64(mu * Float64(Float64(Float64(Float64(Float64(EAccept / KbT) + 2.0) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) / mu) + Float64(-1.0 / KbT)))));
	elseif ((NaChar <= -1.55e-46) || !(NaChar <= 1.6e-26))
		tmp = Float64(t_1 + Float64(-1.0 / Float64(Float64(Float64(Float64(Ec - Float64(EDonor + Float64(mu + Vef))) / KbT) - 2.0) / NdChar)));
	else
		tmp = Float64(t_0 - Float64(NaChar / Float64(Float64(mu - Float64(Float64(Ev + EAccept) + Float64(Vef + Float64(KbT * 2.0)))) / KbT)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)));
	t_1 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	tmp = 0.0;
	if (NaChar <= -3.8e+196)
		tmp = t_1 + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))));
	elseif (NaChar <= -3.5e+57)
		tmp = t_0 + (NaChar / (mu * (((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) / mu) + (-1.0 / KbT))));
	elseif ((NaChar <= -1.55e-46) || ~((NaChar <= 1.6e-26)))
		tmp = t_1 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	else
		tmp = t_0 - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -3.8e+196], N[(t$95$1 + N[(KbT * N[(NdChar / N[(EDonor - N[(Ec - N[(mu + Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, -3.5e+57], N[(t$95$0 + N[(NaChar / N[(mu * N[(N[(N[(N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[NaChar, -1.55e-46], N[Not[LessEqual[NaChar, 1.6e-26]], $MachinePrecision]], N[(t$95$1 + N[(-1.0 / N[(N[(N[(N[(Ec - N[(EDonor + N[(mu + Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision] - 2.0), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(NaChar / N[(N[(mu - N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef + N[(KbT * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -3.8000000000000001e196

   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 54.9%

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

      1. associate-+r+ 54.9%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in KbT around 0 55.0%

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

      1. associate-/l* 61.3%

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

      2. associate--l+ 61.3%

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

      3. +-commutative 61.3%

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

   10. Simplified 61.3%

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

   if -3.8000000000000001e196 < NaChar < -3.4999999999999997e57

   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 58.2%

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

      1. associate-+r+ 58.2%

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

      2. +-commutative 58.2%

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

   7. Simplified 58.2%

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

   8. Taylor expanded in mu around -inf 70.7%

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

      1. associate-*r* 70.7%

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

      2. mul-1-neg 70.7%

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

      3. +-commutative 70.7%

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

      4. mul-1-neg 70.7%

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

      5. unsub-neg 70.7%

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

      6. associate-+r+ 70.7%

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

   10. Simplified 70.7%

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

   if -3.4999999999999997e57 < NaChar < -1.55e-46 or 1.6000000000000001e-26 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 67.2%

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

      1. associate-+r+ 67.2%

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

   7. Simplified 67.2%

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

      1. clear-num 67.8%

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

      2. inv-pow 67.8%

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

      3. associate-+r+ 67.8%

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

   9. Applied egg-rr 67.8%

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

       1. unpow-1 67.8%

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

       2. associate-+l+ 67.8%

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

       3. associate-+r+ 67.8%

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

       4. associate--l+ 67.8%

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

       5. +-commutative 67.8%

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

   11. Simplified 67.8%

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

   12. Taylor expanded in KbT around 0 76.0%

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

   if -1.55e-46 < NaChar < 1.6000000000000001e-26

   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 75.0%

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

      1. associate-+r+ 75.0%

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

      2. +-commutative 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in KbT around 0 80.6%

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

      1. associate-+r+ 80.6%

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

   10. Simplified 80.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -3.8 \cdot 10^{+196}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + KbT \cdot \frac{NdChar}{EDonor - \left(Ec - \left(mu + Vef\right)\right)}\\ \mathbf{elif}\;NaChar \leq -3.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{mu \cdot \left(\frac{\left(\frac{EAccept}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -1.55 \cdot 10^{-46} \lor \neg \left(NaChar \leq 1.6 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{-1}{\frac{\frac{Ec - \left(EDonor + \left(mu + Vef\right)\right)}{KbT} - 2}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} - \frac{NaChar}{\frac{mu - \left(\left(Ev + EAccept\right) + \left(Vef + KbT \cdot 2\right)\right)}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+22} \lor \neg \left(NdChar \leq 1.25 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{-1}{\frac{\frac{Ec - \left(EDonor + \left(mu + Vef\right)\right)}{KbT} - 2}{NdChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -1.6e+22) (not (<= NdChar 1.25e+171)))
   (+
    (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
    (/ NaChar 2.0))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))
    (/ -1.0 (/ (- (/ (- Ec (+ EDonor (+ mu Vef))) KbT) 2.0) NdChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.6e+22) || !(NdChar <= 1.25e+171)) {
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-1.6d+22)) .or. (.not. (ndchar <= 1.25d+171))) then
        tmp = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))) + ((-1.0d0) / ((((ec - (edonor + (mu + vef))) / kbt) - 2.0d0) / ndchar))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.6e+22) || !(NdChar <= 1.25e+171)) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -1.6e+22) or not (NdChar <= 1.25e+171):
		tmp = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -1.6e+22) || !(NdChar <= 1.25e+171))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT)))) + Float64(-1.0 / Float64(Float64(Float64(Float64(Ec - Float64(EDonor + Float64(mu + Vef))) / KbT) - 2.0) / NdChar)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -1.6e+22) || ~((NdChar <= 1.25e+171)))
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -1.6e+22], N[Not[LessEqual[NdChar, 1.25e+171]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[(N[(N[(Ec - N[(EDonor + N[(mu + Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision] - 2.0), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.6e22 or 1.2500000000000001e171 < 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 Ev around inf 87.8%

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

   6. Taylor expanded in Ev around 0 72.8%

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

   if -1.6e22 < NdChar < 1.2500000000000001e171

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 63.5%

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

      1. associate-+r+ 63.5%

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

   7. Simplified 63.5%

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

      1. clear-num 63.8%

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

      2. inv-pow 63.8%

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

      3. associate-+r+ 63.8%

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

   9. Applied egg-rr 63.8%

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

       1. unpow-1 63.8%

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

       2. associate-+l+ 63.8%

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

       3. associate-+r+ 63.8%

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

       4. associate--l+ 63.8%

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

       5. +-commutative 63.8%

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

   11. Simplified 63.8%

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

   12. Taylor expanded in KbT around 0 71.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+22} \lor \neg \left(NdChar \leq 1.25 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{-1}{\frac{\frac{Ec - \left(EDonor + \left(mu + Vef\right)\right)}{KbT} - 2}{NdChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -2.65 \cdot 10^{+170}:\\ \;\;\;\;t\_0 + \frac{NdChar}{2}\\ \mathbf{elif}\;NaChar \leq 8.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} - \frac{NaChar}{\frac{mu - \left(\left(Ev + EAccept\right) + \left(Vef + KbT \cdot 2\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{-1}{\frac{\frac{Ec - \left(EDonor + \left(mu + Vef\right)\right)}{KbT} - 2}{NdChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))))
   (if (<= NaChar -2.65e+170)
     (+ t_0 (/ NdChar 2.0))
     (if (<= NaChar 8.5e-27)
       (-
        (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
        (/ NaChar (/ (- mu (+ (+ Ev EAccept) (+ Vef (* KbT 2.0)))) KbT)))
       (+
        t_0
        (/ -1.0 (/ (- (/ (- Ec (+ EDonor (+ mu Vef))) KbT) 2.0) NdChar)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (NaChar <= -2.65e+170) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NaChar <= 8.5e-27) {
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT));
	} else {
		tmp = t_0 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))
    if (nachar <= (-2.65d+170)) then
        tmp = t_0 + (ndchar / 2.0d0)
    else if (nachar <= 8.5d-27) then
        tmp = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) - (nachar / ((mu - ((ev + eaccept) + (vef + (kbt * 2.0d0)))) / kbt))
    else
        tmp = t_0 + ((-1.0d0) / ((((ec - (edonor + (mu + vef))) / kbt) - 2.0d0) / ndchar))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	double tmp;
	if (NaChar <= -2.65e+170) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NaChar <= 8.5e-27) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT));
	} else {
		tmp = t_0 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))
	tmp = 0
	if NaChar <= -2.65e+170:
		tmp = t_0 + (NdChar / 2.0)
	elif NaChar <= 8.5e-27:
		tmp = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT))
	else:
		tmp = t_0 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT))))
	tmp = 0.0
	if (NaChar <= -2.65e+170)
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	elseif (NaChar <= 8.5e-27)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) - Float64(NaChar / Float64(Float64(mu - Float64(Float64(Ev + EAccept) + Float64(Vef + Float64(KbT * 2.0)))) / KbT)));
	else
		tmp = Float64(t_0 + Float64(-1.0 / Float64(Float64(Float64(Float64(Ec - Float64(EDonor + Float64(mu + Vef))) / KbT) - 2.0) / NdChar)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)));
	tmp = 0.0;
	if (NaChar <= -2.65e+170)
		tmp = t_0 + (NdChar / 2.0);
	elseif (NaChar <= 8.5e-27)
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) - (NaChar / ((mu - ((Ev + EAccept) + (Vef + (KbT * 2.0)))) / KbT));
	else
		tmp = t_0 + (-1.0 / ((((Ec - (EDonor + (mu + Vef))) / KbT) - 2.0) / NdChar));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -2.65e+170], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 8.5e-27], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(N[(mu - N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef + N[(KbT * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.0 / N[(N[(N[(N[(Ec - N[(EDonor + N[(mu + Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision] - 2.0), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -2.65000000000000002e170

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.4%

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

   if -2.65000000000000002e170 < NaChar < 8.50000000000000033e-27

   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 69.8%

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

      1. associate-+r+ 69.8%

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

      2. +-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in KbT around 0 76.2%

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

      1. associate-+r+ 76.2%

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

   10. Simplified 76.2%

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

   if 8.50000000000000033e-27 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 70.7%

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

      1. associate-+r+ 70.7%

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

   7. Simplified 70.7%

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

      1. clear-num 71.4%

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

      2. inv-pow 71.4%

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

      3. associate-+r+ 71.4%

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

   9. Applied egg-rr 71.4%

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

       1. unpow-1 71.4%

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

       2. associate-+l+ 71.4%

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

       3. associate-+r+ 71.4%

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

       4. associate--l+ 71.4%

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

       5. +-commutative 71.4%

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

   11. Simplified 71.4%

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

   12. Taylor expanded in KbT around 0 76.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.65 \cdot 10^{+170}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NaChar \leq 8.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} - \frac{NaChar}{\frac{mu - \left(\left(Ev + EAccept\right) + \left(Vef + KbT \cdot 2\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{-1}{\frac{\frac{Ec - \left(EDonor + \left(mu + Vef\right)\right)}{KbT} - 2}{NdChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.8 \cdot 10^{+22} \lor \neg \left(NdChar \leq 9.6 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + KbT \cdot \frac{NdChar}{EDonor - \left(Ec - \left(mu + Vef\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -1.8e+22) (not (<= NdChar 9.6e-91)))
   (+
    (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
    (/ NaChar 2.0))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))
    (* KbT (/ NdChar (- EDonor (- Ec (+ mu Vef))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.8e+22) || !(NdChar <= 9.6e-91)) {
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-1.8d+22)) .or. (.not. (ndchar <= 9.6d-91))) then
        tmp = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))) + (kbt * (ndchar / (edonor - (ec - (mu + vef)))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.8e+22) || !(NdChar <= 9.6e-91)) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -1.8e+22) or not (NdChar <= 9.6e-91):
		tmp = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -1.8e+22) || !(NdChar <= 9.6e-91))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT)))) + Float64(KbT * Float64(NdChar / Float64(EDonor - Float64(Ec - Float64(mu + Vef))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -1.8e+22) || ~((NdChar <= 9.6e-91)))
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (KbT * (NdChar / (EDonor - (Ec - (mu + Vef)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -1.8e+22], N[Not[LessEqual[NdChar, 9.6e-91]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(KbT * N[(NdChar / N[(EDonor - N[(Ec - N[(mu + Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.8e22 or 9.60000000000000043e-91 < 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 Ev around inf 82.0%

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

   6. Taylor expanded in Ev around 0 66.3%

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

   if -1.8e22 < NdChar < 9.60000000000000043e-91

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 65.4%

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

      1. associate-+r+ 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in KbT around 0 67.5%

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

      1. associate-/l* 72.4%

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

      2. associate--l+ 72.4%

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

      3. +-commutative 72.4%

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

   10. Simplified 72.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.8 \cdot 10^{+22} \lor \neg \left(NdChar \leq 9.6 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + KbT \cdot \frac{NdChar}{EDonor - \left(Ec - \left(mu + Vef\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -6.8 \cdot 10^{+70} \lor \neg \left(NdChar \leq 1.25 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -6.8e+70) (not (<= NdChar 1.25e+171)))
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* NaChar 0.5))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))
    (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -6.8e+70) || !(NdChar <= 1.25e+171)) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-6.8d+70)) .or. (.not. (ndchar <= 1.25d+171))) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -6.8e+70) || !(NdChar <= 1.25e+171)) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -6.8e+70) or not (NdChar <= 1.25e+171):
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -6.8e+70) || !(NdChar <= 1.25e+171))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -6.8e+70) || ~((NdChar <= 1.25e+171)))
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -6.8e+70], N[Not[LessEqual[NdChar, 1.25e+171]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -6.8000000000000002e70 or 1.2500000000000001e171 < 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 mu around inf 66.1%

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

   6. Taylor expanded in KbT around inf 56.4%

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

   if -6.8000000000000002e70 < NdChar < 1.2500000000000001e171

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.9%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -6.8 \cdot 10^{+70} \lor \neg \left(NdChar \leq 1.25 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -7.2 \cdot 10^{+107} \lor \neg \left(NaChar \leq 2.65 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -7.2e+107) (not (<= NaChar 2.65e-33)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (- Ev (- mu EAccept))) KbT))))
    (/ NdChar 2.0))
   (+
    (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
    (/ NaChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -7.2e+107) || !(NaChar <= 2.65e-33)) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((nachar <= (-7.2d+107)) .or. (.not. (nachar <= 2.65d-33))) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev - (mu - eaccept))) / kbt)))) + (ndchar / 2.0d0)
    else
        tmp = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) + (nachar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -7.2e+107) || !(NaChar <= 2.65e-33)) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -7.2e+107) or not (NaChar <= 2.65e-33):
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / 2.0)
	else:
		tmp = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -7.2e+107) || !(NaChar <= 2.65e-33))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev - Float64(mu - EAccept))) / KbT)))) + Float64(NdChar / 2.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) + Float64(NaChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -7.2e+107) || ~((NaChar <= 2.65e-33)))
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev - (mu - EAccept))) / KbT)))) + (NdChar / 2.0);
	else
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -7.2e+107], N[Not[LessEqual[NaChar, 2.65e-33]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev - N[(mu - EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -7.1999999999999995e107 or 2.64999999999999984e-33 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 59.0%

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

   if -7.1999999999999995e107 < NaChar < 2.64999999999999984e-33

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 82.1%

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

   6. Taylor expanded in Ev around 0 66.3%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -7.2 \cdot 10^{+107} \lor \neg \left(NaChar \leq 2.65 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{+71} \lor \neg \left(NdChar \leq 1.6 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -2.4e+71) (not (<= NdChar 1.6e+164)))
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* NaChar 0.5))
   (+ (/ NaChar (+ 1.0 (exp (/ (- (+ Vef Ev) mu) KbT)))) (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.4e+71) || !(NdChar <= 1.6e+164)) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-2.4d+71)) .or. (.not. (ndchar <= 1.6d+164))) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (nachar / (1.0d0 + exp((((vef + ev) - mu) / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.4e+71) || !(NdChar <= 1.6e+164)) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + Ev) - mu) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -2.4e+71) or not (NdChar <= 1.6e+164):
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NaChar / (1.0 + math.exp((((Vef + Ev) - mu) / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -2.4e+71) || !(NdChar <= 1.6e+164))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -2.4e+71) || ~((NdChar <= 1.6e+164)))
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -2.4e+71], N[Not[LessEqual[NdChar, 1.6e+164]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.39999999999999981e71 or 1.5999999999999999e164 < 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 mu around inf 67.7%

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

   6. Taylor expanded in KbT around inf 56.5%

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

   if -2.39999999999999981e71 < NdChar < 1.5999999999999999e164

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.4%

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

   6. Taylor expanded in EAccept around 0 52.8%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{+71} \lor \neg \left(NdChar \leq 1.6 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.5 \cdot 10^{-24} \lor \neg \left(NdChar \leq 1.25 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} - \frac{NaChar}{-1 - e^{\frac{mu}{-KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -1.5e-24) (not (<= NdChar 1.25e+171)))
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* NaChar 0.5))
   (- (/ NdChar 2.0) (/ NaChar (- -1.0 (exp (/ 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 ((NdChar <= -1.5e-24) || !(NdChar <= 1.25e+171)) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - exp((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 ((ndchar <= (-1.5d-24)) .or. (.not. (ndchar <= 1.25d+171))) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (ndchar / 2.0d0) - (nachar / ((-1.0d0) - exp((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 ((NdChar <= -1.5e-24) || !(NdChar <= 1.25e+171)) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - Math.exp((mu / -KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -1.5e-24) or not (NdChar <= 1.25e+171):
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - math.exp((mu / -KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -1.5e-24) || !(NdChar <= 1.25e+171))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NdChar / 2.0) - Float64(NaChar / Float64(-1.0 - exp(Float64(mu / Float64(-KbT))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -1.5e-24) || ~((NdChar <= 1.25e+171)))
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - exp((mu / -KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -1.5e-24], N[Not[LessEqual[NdChar, 1.25e+171]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.49999999999999998e-24 or 1.2500000000000001e171 < 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 mu around inf 64.8%

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

   6. Taylor expanded in KbT around inf 50.5%

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

   if -1.49999999999999998e-24 < NdChar < 1.2500000000000001e171

   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 58.6%

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

   6. Taylor expanded in mu around inf 45.6%

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

      1. associate-*r/ 45.6%

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

      2. mul-1-neg 45.6%

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

   8. Simplified 45.6%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.5 \cdot 10^{-24} \lor \neg \left(NdChar \leq 1.25 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} - \frac{NaChar}{-1 - e^{\frac{mu}{-KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -1.55 \cdot 10^{+121} \lor \neg \left(Vef \leq 1.46 \cdot 10^{+44}\right):\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= Vef -1.55e+121) (not (<= Vef 1.46e+44)))
   (- (* NaChar 0.5) (/ NdChar (- -1.0 (exp (/ Vef KbT)))))
   (- (/ NdChar 2.0) (/ NaChar (- -1.0 (exp (/ Ev KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((Vef <= -1.55e+121) || !(Vef <= 1.46e+44)) {
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - exp((Vef / KbT))));
	} else {
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - exp((Ev / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((vef <= (-1.55d+121)) .or. (.not. (vef <= 1.46d+44))) then
        tmp = (nachar * 0.5d0) - (ndchar / ((-1.0d0) - exp((vef / kbt))))
    else
        tmp = (ndchar / 2.0d0) - (nachar / ((-1.0d0) - exp((ev / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((Vef <= -1.55e+121) || !(Vef <= 1.46e+44)) {
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - Math.exp((Vef / KbT))));
	} else {
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - Math.exp((Ev / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (Vef <= -1.55e+121) or not (Vef <= 1.46e+44):
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - math.exp((Vef / KbT))))
	else:
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - math.exp((Ev / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((Vef <= -1.55e+121) || !(Vef <= 1.46e+44))
		tmp = Float64(Float64(NaChar * 0.5) - Float64(NdChar / Float64(-1.0 - exp(Float64(Vef / KbT)))));
	else
		tmp = Float64(Float64(NdChar / 2.0) - Float64(NaChar / Float64(-1.0 - exp(Float64(Ev / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((Vef <= -1.55e+121) || ~((Vef <= 1.46e+44)))
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - exp((Vef / KbT))));
	else
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - exp((Ev / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[Vef, -1.55e+121], N[Not[LessEqual[Vef, 1.46e+44]], $MachinePrecision]], N[(N[(NaChar * 0.5), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Vef < -1.55000000000000004e121 or 1.4599999999999999e44 < 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 86.0%

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

   6. Taylor expanded in KbT around inf 40.0%

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

   if -1.55000000000000004e121 < Vef < 1.4599999999999999e44

   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 57.8%

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

   6. Taylor expanded in Ev around inf 47.8%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.55 \cdot 10^{+121} \lor \neg \left(Vef \leq 1.46 \cdot 10^{+44}\right):\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -9.2 \cdot 10^{+70} \lor \neg \left(NdChar \leq 7.2 \cdot 10^{+153}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -9.2e+70) (not (<= NdChar 7.2e+153)))
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* NaChar 0.5))
   (- (/ NdChar 2.0) (/ NaChar (- -1.0 (exp (/ Ev KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -9.2e+70) || !(NdChar <= 7.2e+153)) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - exp((Ev / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-9.2d+70)) .or. (.not. (ndchar <= 7.2d+153))) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (ndchar / 2.0d0) - (nachar / ((-1.0d0) - exp((ev / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -9.2e+70) || !(NdChar <= 7.2e+153)) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - Math.exp((Ev / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -9.2e+70) or not (NdChar <= 7.2e+153):
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - math.exp((Ev / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -9.2e+70) || !(NdChar <= 7.2e+153))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NdChar / 2.0) - Float64(NaChar / Float64(-1.0 - exp(Float64(Ev / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -9.2e+70) || ~((NdChar <= 7.2e+153)))
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - exp((Ev / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -9.2e+70], N[Not[LessEqual[NdChar, 7.2e+153]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -9.19999999999999975e70 or 7.2000000000000001e153 < 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 mu around inf 67.6%

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

   6. Taylor expanded in KbT around inf 54.9%

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

   if -9.19999999999999975e70 < NdChar < 7.2000000000000001e153

   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 57.2%

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

   6. Taylor expanded in Ev around inf 40.9%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -9.2 \cdot 10^{+70} \lor \neg \left(NdChar \leq 7.2 \cdot 10^{+153}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -4.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{2} - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -4.7e+21)
   (- (/ NdChar 2.0) (/ NaChar (- -1.0 (exp (/ Ev KbT)))))
   (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -4.7e+21) {
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - exp((Ev / KbT))));
	} else {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ev <= (-4.7d+21)) then
        tmp = (ndchar / 2.0d0) - (nachar / ((-1.0d0) - exp((ev / kbt))))
    else
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -4.7e+21) {
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - Math.exp((Ev / KbT))));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -4.7e+21:
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - math.exp((Ev / KbT))))
	else:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -4.7e+21)
		tmp = Float64(Float64(NdChar / 2.0) - Float64(NaChar / Float64(-1.0 - exp(Float64(Ev / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -4.7e+21)
		tmp = (NdChar / 2.0) - (NaChar / (-1.0 - exp((Ev / KbT))));
	else
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -4.7e+21], N[(N[(NdChar / 2.0), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Ev < -4.7e21

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.1%

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

   6. Taylor expanded in Ev around inf 47.0%

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

   if -4.7e21 < Ev

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 50.4%

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

   6. Taylor expanded in EAccept around inf 41.7%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -4.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{2} - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (/ NdChar 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 / (1.0 + exp((EAccept / KbT)))) + (NdChar / 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 / (1.0d0 + exp((eaccept / kbt)))) + (ndchar / 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 / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar / 2.0);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar / 2.0)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar / 2.0))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in KbT around inf 50.8%

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

6. Taylor expanded in EAccept around inf 41.3%

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

7. Final simplification 41.3%

   \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2} \]
8. Add Preprocessing

Alternative 20: 27.6% accurate, 32.7× speedup

Math

\[\begin{array}{l} \\ \frac{NdChar}{2} + NaChar \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+ (/ NdChar 2.0) (* NaChar 0.5)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / 2.0) + (NaChar * 0.5);
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (ndchar / 2.0d0) + (nachar * 0.5d0)
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / 2.0) + (NaChar * 0.5);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / 2.0) + (NaChar * 0.5)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / 2.0) + Float64(NaChar * 0.5))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / 2.0) + (NaChar * 0.5);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in KbT around inf 50.8%

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

6. Taylor expanded in KbT around inf 33.1%

   \[\leadsto \frac{NdChar}{2} + \color{blue}{0.5 \cdot NaChar} \]

7. Final simplification 33.1%

   \[\leadsto \frac{NdChar}{2} + NaChar \cdot 0.5 \]
8. Add Preprocessing

Alternative 21: 18.3% accurate, 76.3× speedup

Math

\[\begin{array}{l} \\ NdChar \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* NdChar 0.5))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NdChar * 0.5;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = ndchar * 0.5d0
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NdChar * 0.5;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return NdChar * 0.5

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(NdChar * 0.5)
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = NdChar * 0.5;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(NdChar * 0.5), $MachinePrecision]

Derivation

1. Initial program 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 57.6%

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

   1. associate-+r+ 57.6%

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

   2. +-commutative 57.6%

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

7. Simplified 57.6%

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

8. Taylor expanded in mu around inf 35.1%

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

   1. neg-mul-1 35.1%

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

   2. distribute-neg-frac2 35.1%

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

10. Simplified 35.1%

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

11. Taylor expanded in KbT around inf 12.9%

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

12. Taylor expanded in NdChar around inf 23.2%

    \[\leadsto \color{blue}{0.5 \cdot NdChar} \]

13. Final simplification 23.2%

    \[\leadsto NdChar \cdot 0.5 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))))))