Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 31.2s

Alternatives: 29

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

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 + pow(((double) M_E), (((Vef + EAccept) + (Ev - mu)) / KbT)))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
}

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.pow(Math.E, (((Vef + EAccept) + (Ev - mu)) / KbT)))) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
}

Python

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

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NaChar / Float64(1.0 + (exp(1) ^ Float64(Float64(Float64(Vef + EAccept) + Float64(Ev - mu)) / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NaChar / (1.0 + (2.71828182845904523536 ^ (((Vef + EAccept) + (Ev - mu)) / KbT)))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
end

Wolfram

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

   1. *-un-lft-identity 100.0%

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

   2. exp-prod 100.0%

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

6. Applied egg-rr 100.0%

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

   1. exp-1-e 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-+l- 100.0%

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

   4. unsub-neg 100.0%

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

   5. mul-1-neg 100.0%

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

   6. associate--l+ 100.0%

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

   7. mul-1-neg 100.0%

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

   8. unsub-neg 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\frac{Vef}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ t_2 := \frac{NdChar}{t\_0} + \frac{NaChar}{t\_0}\\ \mathbf{if}\;Vef \leq -1.15 \cdot 10^{+139}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Vef \leq -1.6 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq -1.65 \cdot 10^{-38}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{elif}\;Vef \leq -6.8 \cdot 10^{-193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq 3.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;Vef \leq 3.7 \cdot 10^{+193}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (/ Vef KbT))))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
          (/ NaChar (- -1.0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT))))))
        (t_2 (+ (/ NdChar t_0) (/ NaChar t_0))))
   (if (<= Vef -1.15e+139)
     t_2
     (if (<= Vef -1.6e+32)
       t_1
       (if (<= Vef -1.65e-38)
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (/
           NaChar
           (-
            1.0
            (*
             mu
             (+
              (/ 1.0 KbT)
              (/
               (- -1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
               mu))))))
         (if (<= Vef -6.8e-193)
           t_1
           (if (<= Vef 3.9e-224)
             (+
              (/ NdChar (+ 1.0 (exp (/ mu KbT))))
              (/ NaChar (+ 1.0 (exp (/ mu (- KbT))))))
             (if (<= Vef 3.7e+193) t_1 t_2))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 1.0 + exp((Vef / KbT));
	double t_1 = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	double t_2 = (NdChar / t_0) + (NaChar / t_0);
	double tmp;
	if (Vef <= -1.15e+139) {
		tmp = t_2;
	} else if (Vef <= -1.6e+32) {
		tmp = t_1;
	} else if (Vef <= -1.65e-38) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else if (Vef <= -6.8e-193) {
		tmp = t_1;
	} else if (Vef <= 3.9e-224) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	} else if (Vef <= 3.7e+193) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + exp((vef / kbt))
    t_1 = (ndchar / (1.0d0 + exp((edonor / kbt)))) - (nachar / ((-1.0d0) - exp(((vef - ((mu - eaccept) - ev)) / kbt))))
    t_2 = (ndchar / t_0) + (nachar / t_0)
    if (vef <= (-1.15d+139)) then
        tmp = t_2
    else if (vef <= (-1.6d+32)) then
        tmp = t_1
    else if (vef <= (-1.65d-38)) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / (1.0d0 - (mu * ((1.0d0 / kbt) + (((-1.0d0) - ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) / mu)))))
    else if (vef <= (-6.8d-193)) then
        tmp = t_1
    else if (vef <= 3.9d-224) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((mu / -kbt))))
    else if (vef <= 3.7d+193) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 1.0 + Math.exp((Vef / KbT));
	double t_1 = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) - (NaChar / (-1.0 - Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	double t_2 = (NdChar / t_0) + (NaChar / t_0);
	double tmp;
	if (Vef <= -1.15e+139) {
		tmp = t_2;
	} else if (Vef <= -1.6e+32) {
		tmp = t_1;
	} else if (Vef <= -1.65e-38) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else if (Vef <= -6.8e-193) {
		tmp = t_1;
	} else if (Vef <= 3.9e-224) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	} else if (Vef <= 3.7e+193) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 / KbT)))) - (NaChar / (-1.0 - math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))))
	t_2 = (NdChar / t_0) + (NaChar / t_0)
	tmp = 0
	if Vef <= -1.15e+139:
		tmp = t_2
	elif Vef <= -1.6e+32:
		tmp = t_1
	elif Vef <= -1.65e-38:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))))
	elif Vef <= -6.8e-193:
		tmp = t_1
	elif Vef <= 3.9e-224:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((mu / -KbT))))
	elif Vef <= 3.7e+193:
		tmp = t_1
	else:
		tmp = t_2
	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(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT)))))
	t_2 = Float64(Float64(NdChar / t_0) + Float64(NaChar / t_0))
	tmp = 0.0
	if (Vef <= -1.15e+139)
		tmp = t_2;
	elseif (Vef <= -1.6e+32)
		tmp = t_1;
	elseif (Vef <= -1.65e-38)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 - Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Float64(-1.0 - Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) / mu))))));
	elseif (Vef <= -6.8e-193)
		tmp = t_1;
	elseif (Vef <= 3.9e-224)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))));
	elseif (Vef <= 3.7e+193)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 1.0 + exp((Vef / KbT));
	t_1 = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	t_2 = (NdChar / t_0) + (NaChar / t_0);
	tmp = 0.0;
	if (Vef <= -1.15e+139)
		tmp = t_2;
	elseif (Vef <= -1.6e+32)
		tmp = t_1;
	elseif (Vef <= -1.65e-38)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	elseif (Vef <= -6.8e-193)
		tmp = t_1;
	elseif (Vef <= 3.9e-224)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	elseif (Vef <= 3.7e+193)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / t$95$0), $MachinePrecision] + N[(NaChar / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -1.15e+139], t$95$2, If[LessEqual[Vef, -1.6e+32], t$95$1, If[LessEqual[Vef, -1.65e-38], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 - N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(-1.0 - N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, -6.8e-193], t$95$1, If[LessEqual[Vef, 3.9e-224], 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[Vef, 3.7e+193], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -1.15e139 or 3.7000000000000003e193 < 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 93.1%

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

   6. Taylor expanded in Vef around inf 85.8%

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

   if -1.15e139 < Vef < -1.5999999999999999e32 or -1.6500000000000001e-38 < Vef < -6.8000000000000004e-193 or 3.8999999999999998e-224 < Vef < 3.7000000000000003e193

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 75.1%

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

   if -1.5999999999999999e32 < Vef < -1.6500000000000001e-38

   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 76.1%

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

      1. +-commutative 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in mu around -inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. mul-1-neg 83.8%

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

      3. +-commutative 83.8%

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

      4. mul-1-neg 83.8%

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

      5. unsub-neg 83.8%

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

   10. Simplified 83.8%

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

   if -6.8000000000000004e-193 < Vef < 3.8999999999999998e-224

   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 86.9%

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

   6. Taylor expanded in mu around inf 86.2%

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

      1. neg-mul-1 86.2%

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

      2. distribute-neg-frac2 86.2%

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

   8. Simplified 86.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.15 \cdot 10^{+139}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq -1.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq -1.65 \cdot 10^{-38}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{elif}\;Vef \leq -6.8 \cdot 10^{-193}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 3.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;Vef \leq 3.7 \cdot 10^{+193}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}\\ t_1 := \frac{NaChar}{-1 - t\_0}\\ t_2 := \frac{NdChar}{1 + e^{\frac{Ec}{-KbT}}} - t\_1\\ t_3 := \frac{NaChar}{1 + t\_0} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Vef \leq -6.9 \cdot 10^{+146}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;Vef \leq -5.4 \cdot 10^{-204}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Vef \leq 4.1 \cdot 10^{-193}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} - t\_1\\ \mathbf{elif}\;Vef \leq 6.2 \cdot 10^{-129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Vef \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT)))
        (t_1 (/ NaChar (- -1.0 t_0)))
        (t_2 (- (/ NdChar (+ 1.0 (exp (/ Ec (- KbT))))) t_1))
        (t_3 (+ (/ NaChar (+ 1.0 t_0)) (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))))
   (if (<= Vef -6.9e+146)
     t_3
     (if (<= Vef -5.4e-204)
       t_2
       (if (<= Vef 4.1e-193)
         (- (/ NdChar (+ 1.0 (exp (/ mu KbT)))) t_1)
         (if (<= Vef 6.2e-129)
           t_2
           (if (<= Vef 2e+113)
             (- (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_1)
             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 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_1 = NaChar / (-1.0 - t_0);
	double t_2 = (NdChar / (1.0 + exp((Ec / -KbT)))) - t_1;
	double t_3 = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + exp((Vef / KbT))));
	double tmp;
	if (Vef <= -6.9e+146) {
		tmp = t_3;
	} else if (Vef <= -5.4e-204) {
		tmp = t_2;
	} else if (Vef <= 4.1e-193) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) - t_1;
	} else if (Vef <= 6.2e-129) {
		tmp = t_2;
	} else if (Vef <= 2e+113) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - t_1;
	} 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 = exp(((vef - ((mu - eaccept) - ev)) / kbt))
    t_1 = nachar / ((-1.0d0) - t_0)
    t_2 = (ndchar / (1.0d0 + exp((ec / -kbt)))) - t_1
    t_3 = (nachar / (1.0d0 + t_0)) + (ndchar / (1.0d0 + exp((vef / kbt))))
    if (vef <= (-6.9d+146)) then
        tmp = t_3
    else if (vef <= (-5.4d-204)) then
        tmp = t_2
    else if (vef <= 4.1d-193) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) - t_1
    else if (vef <= 6.2d-129) then
        tmp = t_2
    else if (vef <= 2d+113) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) - t_1
    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 = Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_1 = NaChar / (-1.0 - t_0);
	double t_2 = (NdChar / (1.0 + Math.exp((Ec / -KbT)))) - t_1;
	double t_3 = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	double tmp;
	if (Vef <= -6.9e+146) {
		tmp = t_3;
	} else if (Vef <= -5.4e-204) {
		tmp = t_2;
	} else if (Vef <= 4.1e-193) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) - t_1;
	} else if (Vef <= 6.2e-129) {
		tmp = t_2;
	} else if (Vef <= 2e+113) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) - t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))
	t_1 = NaChar / (-1.0 - t_0)
	t_2 = (NdChar / (1.0 + math.exp((Ec / -KbT)))) - t_1
	t_3 = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + math.exp((Vef / KbT))))
	tmp = 0
	if Vef <= -6.9e+146:
		tmp = t_3
	elif Vef <= -5.4e-204:
		tmp = t_2
	elif Vef <= 4.1e-193:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) - t_1
	elif Vef <= 6.2e-129:
		tmp = t_2
	elif Vef <= 2e+113:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) - t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT))
	t_1 = Float64(NaChar / Float64(-1.0 - t_0))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Ec / Float64(-KbT))))) - t_1)
	t_3 = Float64(Float64(NaChar / Float64(1.0 + t_0)) + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))))
	tmp = 0.0
	if (Vef <= -6.9e+146)
		tmp = t_3;
	elseif (Vef <= -5.4e-204)
		tmp = t_2;
	elseif (Vef <= 4.1e-193)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) - t_1);
	elseif (Vef <= 6.2e-129)
		tmp = t_2;
	elseif (Vef <= 2e+113)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) - t_1);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	t_1 = NaChar / (-1.0 - t_0);
	t_2 = (NdChar / (1.0 + exp((Ec / -KbT)))) - t_1;
	t_3 = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + exp((Vef / KbT))));
	tmp = 0.0;
	if (Vef <= -6.9e+146)
		tmp = t_3;
	elseif (Vef <= -5.4e-204)
		tmp = t_2;
	elseif (Vef <= 4.1e-193)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) - t_1;
	elseif (Vef <= 6.2e-129)
		tmp = t_2;
	elseif (Vef <= 2e+113)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - t_1;
	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[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -6.9e+146], t$95$3, If[LessEqual[Vef, -5.4e-204], t$95$2, If[LessEqual[Vef, 4.1e-193], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[Vef, 6.2e-129], t$95$2, If[LessEqual[Vef, 2e+113], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -6.89999999999999956e146 or 2e113 < 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 92.9%

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

   if -6.89999999999999956e146 < Vef < -5.39999999999999983e-204 or 4.10000000000000003e-193 < Vef < 6.2000000000000001e-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 Ec around inf 79.1%

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

      1. associate-*r/ 79.1%

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

      2. mul-1-neg 79.1%

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

   7. Simplified 79.1%

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

   if -5.39999999999999983e-204 < Vef < 4.10000000000000003e-193

   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 85.3%

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

   if 6.2000000000000001e-129 < Vef < 2e113

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 75.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -6.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq -5.4 \cdot 10^{-204}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Ec}{-KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 4.1 \cdot 10^{-193}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 6.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Ec}{-KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}\\ t_1 := e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}\\ t_2 := \frac{NaChar}{-1 - t\_1}\\ t_3 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} - t\_2\\ \mathbf{if}\;mu \leq -4.4 \cdot 10^{+95}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;mu \leq -6.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{NaChar}{1 + t\_1} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq 7.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{NdChar}{1 + t\_0} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{elif}\;mu \leq 850000000000:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - t\_2\\ \mathbf{elif}\;mu \leq 8 \cdot 10^{+48}:\\ \;\;\;\;\frac{NaChar}{\left(2 - Ev \cdot \left(\frac{-1}{KbT} - \frac{Vef}{KbT \cdot Ev}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))
        (t_1 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT)))
        (t_2 (/ NaChar (- -1.0 t_1)))
        (t_3 (- (/ NdChar (+ 1.0 (exp (/ mu KbT)))) t_2)))
   (if (<= mu -4.4e+95)
     t_3
     (if (<= mu -6.6e-42)
       (+ (/ NaChar (+ 1.0 t_1)) (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
       (if (<= mu 7.5e-130)
         (+
          (/ NdChar (+ 1.0 t_0))
          (/
           NaChar
           (-
            1.0
            (*
             mu
             (+
              (/ 1.0 KbT)
              (/
               (- -1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
               mu))))))
         (if (<= mu 850000000000.0)
           (- (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_2)
           (if (<= mu 8e+48)
             (-
              (/
               NaChar
               (-
                (- 2.0 (* Ev (- (/ -1.0 KbT) (/ Vef (* KbT Ev)))))
                (/ mu KbT)))
              (/ NdChar (- -1.0 t_0)))
             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 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_2 = NaChar / (-1.0 - t_1);
	double t_3 = (NdChar / (1.0 + exp((mu / KbT)))) - t_2;
	double tmp;
	if (mu <= -4.4e+95) {
		tmp = t_3;
	} else if (mu <= -6.6e-42) {
		tmp = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + exp((Vef / KbT))));
	} else if (mu <= 7.5e-130) {
		tmp = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else if (mu <= 850000000000.0) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - t_2;
	} else if (mu <= 8e+48) {
		tmp = (NaChar / ((2.0 - (Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	} 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 = exp(((edonor + (mu + (vef - ec))) / kbt))
    t_1 = exp(((vef - ((mu - eaccept) - ev)) / kbt))
    t_2 = nachar / ((-1.0d0) - t_1)
    t_3 = (ndchar / (1.0d0 + exp((mu / kbt)))) - t_2
    if (mu <= (-4.4d+95)) then
        tmp = t_3
    else if (mu <= (-6.6d-42)) then
        tmp = (nachar / (1.0d0 + t_1)) + (ndchar / (1.0d0 + exp((vef / kbt))))
    else if (mu <= 7.5d-130) then
        tmp = (ndchar / (1.0d0 + t_0)) + (nachar / (1.0d0 - (mu * ((1.0d0 / kbt) + (((-1.0d0) - ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) / mu)))))
    else if (mu <= 850000000000.0d0) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) - t_2
    else if (mu <= 8d+48) then
        tmp = (nachar / ((2.0d0 - (ev * (((-1.0d0) / kbt) - (vef / (kbt * ev))))) - (mu / kbt))) - (ndchar / ((-1.0d0) - t_0))
    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 = Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_2 = NaChar / (-1.0 - t_1);
	double t_3 = (NdChar / (1.0 + Math.exp((mu / KbT)))) - t_2;
	double tmp;
	if (mu <= -4.4e+95) {
		tmp = t_3;
	} else if (mu <= -6.6e-42) {
		tmp = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	} else if (mu <= 7.5e-130) {
		tmp = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else if (mu <= 850000000000.0) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) - t_2;
	} else if (mu <= 8e+48) {
		tmp = (NaChar / ((2.0 - (Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))
	t_1 = math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))
	t_2 = NaChar / (-1.0 - t_1)
	t_3 = (NdChar / (1.0 + math.exp((mu / KbT)))) - t_2
	tmp = 0
	if mu <= -4.4e+95:
		tmp = t_3
	elif mu <= -6.6e-42:
		tmp = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + math.exp((Vef / KbT))))
	elif mu <= 7.5e-130:
		tmp = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))))
	elif mu <= 850000000000.0:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) - t_2
	elif mu <= 8e+48:
		tmp = (NaChar / ((2.0 - (Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0))
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))
	t_1 = exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT))
	t_2 = Float64(NaChar / Float64(-1.0 - t_1))
	t_3 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) - t_2)
	tmp = 0.0
	if (mu <= -4.4e+95)
		tmp = t_3;
	elseif (mu <= -6.6e-42)
		tmp = Float64(Float64(NaChar / Float64(1.0 + t_1)) + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	elseif (mu <= 7.5e-130)
		tmp = Float64(Float64(NdChar / Float64(1.0 + t_0)) + Float64(NaChar / Float64(1.0 - Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Float64(-1.0 - Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) / mu))))));
	elseif (mu <= 850000000000.0)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) - t_2);
	elseif (mu <= 8e+48)
		tmp = Float64(Float64(NaChar / Float64(Float64(2.0 - Float64(Ev * Float64(Float64(-1.0 / KbT) - Float64(Vef / Float64(KbT * Ev))))) - Float64(mu / KbT))) - Float64(NdChar / Float64(-1.0 - t_0)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	t_1 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	t_2 = NaChar / (-1.0 - t_1);
	t_3 = (NdChar / (1.0 + exp((mu / KbT)))) - t_2;
	tmp = 0.0;
	if (mu <= -4.4e+95)
		tmp = t_3;
	elseif (mu <= -6.6e-42)
		tmp = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + exp((Vef / KbT))));
	elseif (mu <= 7.5e-130)
		tmp = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	elseif (mu <= 850000000000.0)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - t_2;
	elseif (mu <= 8e+48)
		tmp = (NaChar / ((2.0 - (Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	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[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[mu, -4.4e+95], t$95$3, If[LessEqual[mu, -6.6e-42], N[(N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 7.5e-130], N[(N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 - N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(-1.0 - N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 850000000000.0], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[mu, 8e+48], N[(N[(NaChar / N[(N[(2.0 - N[(Ev * N[(N[(-1.0 / KbT), $MachinePrecision] - N[(Vef / N[(KbT * Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if mu < -4.3999999999999998e95 or 8.00000000000000035e48 < 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 84.3%

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

   if -4.3999999999999998e95 < mu < -6.6000000000000005e-42

   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 74.1%

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

   if -6.6000000000000005e-42 < mu < 7.4999999999999994e-130

   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 63.2%

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

      1. +-commutative 63.2%

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

   7. Simplified 63.2%

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

   8. Taylor expanded in mu around -inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. mul-1-neg 77.7%

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

      3. +-commutative 77.7%

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

      4. mul-1-neg 77.7%

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

      5. unsub-neg 77.7%

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

   10. Simplified 77.7%

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

   if 7.4999999999999994e-130 < mu < 8.5e11

   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 EDonor around inf 80.4%

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

   if 8.5e11 < mu < 8.00000000000000035e48

   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 80.4%

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

      1. +-commutative 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in Ev around inf 90.4%

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

   9. Taylor expanded in EAccept around 0 81.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -4.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;mu \leq -6.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq 7.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{elif}\;mu \leq 850000000000:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;mu \leq 8 \cdot 10^{+48}:\\ \;\;\;\;\frac{NaChar}{\left(2 - Ev \cdot \left(\frac{-1}{KbT} - \frac{Vef}{KbT \cdot Ev}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}\\ t_1 := e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}\\ t_2 := \frac{NaChar}{1 + t\_1} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ t_3 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - t\_1}\\ \mathbf{if}\;EDonor \leq -4.2 \cdot 10^{+118}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;EDonor \leq 2 \cdot 10^{-233}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;EDonor \leq 4 \cdot 10^{-99}:\\ \;\;\;\;\frac{NdChar}{1 + t\_0} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{elif}\;EDonor \leq 6 \cdot 10^{-72}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;EDonor \leq 1.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 - \frac{Ev \cdot \left(-1 - \frac{Vef}{Ev}\right) - EAccept}{KbT}\right) - \frac{mu}{KbT}\right)} - \frac{NdChar}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))
        (t_1 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT)))
        (t_2 (+ (/ NaChar (+ 1.0 t_1)) (/ NdChar (+ 1.0 (exp (/ Vef KbT))))))
        (t_3
         (- (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (/ NaChar (- -1.0 t_1)))))
   (if (<= EDonor -4.2e+118)
     t_3
     (if (<= EDonor 2e-233)
       t_2
       (if (<= EDonor 4e-99)
         (+
          (/ NdChar (+ 1.0 t_0))
          (/
           NaChar
           (-
            1.0
            (*
             mu
             (+
              (/ 1.0 KbT)
              (/
               (- -1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
               mu))))))
         (if (<= EDonor 6e-72)
           t_2
           (if (<= EDonor 1.1e+23)
             (-
              (/
               NaChar
               (+
                1.0
                (-
                 (- 1.0 (/ (- (* Ev (- -1.0 (/ Vef Ev))) EAccept) KbT))
                 (/ mu KbT))))
              (/ NdChar (- -1.0 t_0)))
             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 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_2 = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + exp((Vef / KbT))));
	double t_3 = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_1));
	double tmp;
	if (EDonor <= -4.2e+118) {
		tmp = t_3;
	} else if (EDonor <= 2e-233) {
		tmp = t_2;
	} else if (EDonor <= 4e-99) {
		tmp = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else if (EDonor <= 6e-72) {
		tmp = t_2;
	} else if (EDonor <= 1.1e+23) {
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - (NdChar / (-1.0 - t_0));
	} 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 = exp(((edonor + (mu + (vef - ec))) / kbt))
    t_1 = exp(((vef - ((mu - eaccept) - ev)) / kbt))
    t_2 = (nachar / (1.0d0 + t_1)) + (ndchar / (1.0d0 + exp((vef / kbt))))
    t_3 = (ndchar / (1.0d0 + exp((edonor / kbt)))) - (nachar / ((-1.0d0) - t_1))
    if (edonor <= (-4.2d+118)) then
        tmp = t_3
    else if (edonor <= 2d-233) then
        tmp = t_2
    else if (edonor <= 4d-99) then
        tmp = (ndchar / (1.0d0 + t_0)) + (nachar / (1.0d0 - (mu * ((1.0d0 / kbt) + (((-1.0d0) - ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) / mu)))))
    else if (edonor <= 6d-72) then
        tmp = t_2
    else if (edonor <= 1.1d+23) then
        tmp = (nachar / (1.0d0 + ((1.0d0 - (((ev * ((-1.0d0) - (vef / ev))) - eaccept) / kbt)) - (mu / kbt)))) - (ndchar / ((-1.0d0) - t_0))
    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 = Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_2 = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	double t_3 = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_1));
	double tmp;
	if (EDonor <= -4.2e+118) {
		tmp = t_3;
	} else if (EDonor <= 2e-233) {
		tmp = t_2;
	} else if (EDonor <= 4e-99) {
		tmp = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else if (EDonor <= 6e-72) {
		tmp = t_2;
	} else if (EDonor <= 1.1e+23) {
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - (NdChar / (-1.0 - t_0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))
	t_1 = math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))
	t_2 = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + math.exp((Vef / KbT))))
	t_3 = (NdChar / (1.0 + math.exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_1))
	tmp = 0
	if EDonor <= -4.2e+118:
		tmp = t_3
	elif EDonor <= 2e-233:
		tmp = t_2
	elif EDonor <= 4e-99:
		tmp = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))))
	elif EDonor <= 6e-72:
		tmp = t_2
	elif EDonor <= 1.1e+23:
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - (NdChar / (-1.0 - t_0))
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))
	t_1 = exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT))
	t_2 = Float64(Float64(NaChar / Float64(1.0 + t_1)) + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))))
	t_3 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) - Float64(NaChar / Float64(-1.0 - t_1)))
	tmp = 0.0
	if (EDonor <= -4.2e+118)
		tmp = t_3;
	elseif (EDonor <= 2e-233)
		tmp = t_2;
	elseif (EDonor <= 4e-99)
		tmp = Float64(Float64(NdChar / Float64(1.0 + t_0)) + Float64(NaChar / Float64(1.0 - Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Float64(-1.0 - Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) / mu))))));
	elseif (EDonor <= 6e-72)
		tmp = t_2;
	elseif (EDonor <= 1.1e+23)
		tmp = Float64(Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 - Float64(Float64(Float64(Ev * Float64(-1.0 - Float64(Vef / Ev))) - EAccept) / KbT)) - Float64(mu / KbT)))) - Float64(NdChar / Float64(-1.0 - t_0)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	t_1 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	t_2 = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + exp((Vef / KbT))));
	t_3 = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_1));
	tmp = 0.0;
	if (EDonor <= -4.2e+118)
		tmp = t_3;
	elseif (EDonor <= 2e-233)
		tmp = t_2;
	elseif (EDonor <= 4e-99)
		tmp = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	elseif (EDonor <= 6e-72)
		tmp = t_2;
	elseif (EDonor <= 1.1e+23)
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - (NdChar / (-1.0 - t_0));
	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[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EDonor, -4.2e+118], t$95$3, If[LessEqual[EDonor, 2e-233], t$95$2, If[LessEqual[EDonor, 4e-99], N[(N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 - N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(-1.0 - N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 6e-72], t$95$2, If[LessEqual[EDonor, 1.1e+23], N[(N[(NaChar / N[(1.0 + N[(N[(1.0 - N[(N[(N[(Ev * N[(-1.0 - N[(Vef / Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if EDonor < -4.2e118 or 1.10000000000000004e23 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 82.6%

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

   if -4.2e118 < EDonor < 1.99999999999999992e-233 or 4.0000000000000001e-99 < EDonor < 6e-72

   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 75.4%

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

   if 1.99999999999999992e-233 < EDonor < 4.0000000000000001e-99

   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 78.1%

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

      1. +-commutative 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in mu around -inf 84.2%

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

      1. associate-*r* 84.2%

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

      2. mul-1-neg 84.2%

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

      3. +-commutative 84.2%

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

      4. mul-1-neg 84.2%

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

      5. unsub-neg 84.2%

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

   10. Simplified 84.2%

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

   if 6e-72 < EDonor < 1.10000000000000004e23

   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 82.6%

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

      1. +-commutative 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in Ev around inf 76.7%

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

   9. Taylor expanded in KbT around 0 88.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -4.2 \cdot 10^{+118}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;EDonor \leq 2 \cdot 10^{-233}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;EDonor \leq 4 \cdot 10^{-99}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{elif}\;EDonor \leq 6 \cdot 10^{-72}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;EDonor \leq 1.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 - \frac{Ev \cdot \left(-1 - \frac{Vef}{Ev}\right) - EAccept}{KbT}\right) - \frac{mu}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\frac{Vef}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ t_2 := \frac{NdChar}{t\_0} + \frac{NaChar}{t\_0}\\ \mathbf{if}\;Vef \leq -9.5 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Vef \leq -4 \cdot 10^{+58}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq -1.85 \cdot 10^{-136}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Vef \leq 8 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (/ Vef KbT))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (/
           NaChar
           (-
            1.0
            (*
             mu
             (+
              (/ 1.0 KbT)
              (/
               (- -1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
               mu)))))))
        (t_2 (+ (/ NdChar t_0) (/ NaChar t_0))))
   (if (<= Vef -9.5e+138)
     t_2
     (if (<= Vef -4e+58)
       (-
        (/ NdChar (+ (/ Vef KbT) 2.0))
        (/ NaChar (- -1.0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT)))))
       (if (<= Vef -2.5e-38)
         t_1
         (if (<= Vef -1.85e-136)
           (+
            (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
            (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
           (if (<= Vef 8e+74) t_1 t_2)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 1.0 + exp((Vef / KbT));
	double t_1 = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	double t_2 = (NdChar / t_0) + (NaChar / t_0);
	double tmp;
	if (Vef <= -9.5e+138) {
		tmp = t_2;
	} else if (Vef <= -4e+58) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	} else if (Vef <= -2.5e-38) {
		tmp = t_1;
	} else if (Vef <= -1.85e-136) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (Vef <= 8e+74) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + exp((vef / kbt))
    t_1 = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / (1.0d0 - (mu * ((1.0d0 / kbt) + (((-1.0d0) - ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) / mu)))))
    t_2 = (ndchar / t_0) + (nachar / t_0)
    if (vef <= (-9.5d+138)) then
        tmp = t_2
    else if (vef <= (-4d+58)) then
        tmp = (ndchar / ((vef / kbt) + 2.0d0)) - (nachar / ((-1.0d0) - exp(((vef - ((mu - eaccept) - ev)) / kbt))))
    else if (vef <= (-2.5d-38)) then
        tmp = t_1
    else if (vef <= (-1.85d-136)) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (vef <= 8d+74) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 1.0 + Math.exp((Vef / KbT));
	double t_1 = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	double t_2 = (NdChar / t_0) + (NaChar / t_0);
	double tmp;
	if (Vef <= -9.5e+138) {
		tmp = t_2;
	} else if (Vef <= -4e+58) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	} else if (Vef <= -2.5e-38) {
		tmp = t_1;
	} else if (Vef <= -1.85e-136) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (Vef <= 8e+74) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))))
	t_2 = (NdChar / t_0) + (NaChar / t_0)
	tmp = 0
	if Vef <= -9.5e+138:
		tmp = t_2
	elif Vef <= -4e+58:
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))))
	elif Vef <= -2.5e-38:
		tmp = t_1
	elif Vef <= -1.85e-136:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif Vef <= 8e+74:
		tmp = t_1
	else:
		tmp = t_2
	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(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 - Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Float64(-1.0 - Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) / mu))))))
	t_2 = Float64(Float64(NdChar / t_0) + Float64(NaChar / t_0))
	tmp = 0.0
	if (Vef <= -9.5e+138)
		tmp = t_2;
	elseif (Vef <= -4e+58)
		tmp = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 2.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT)))));
	elseif (Vef <= -2.5e-38)
		tmp = t_1;
	elseif (Vef <= -1.85e-136)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (Vef <= 8e+74)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 1.0 + exp((Vef / KbT));
	t_1 = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	t_2 = (NdChar / t_0) + (NaChar / t_0);
	tmp = 0.0;
	if (Vef <= -9.5e+138)
		tmp = t_2;
	elseif (Vef <= -4e+58)
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	elseif (Vef <= -2.5e-38)
		tmp = t_1;
	elseif (Vef <= -1.85e-136)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (Vef <= 8e+74)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 - N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(-1.0 - N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / t$95$0), $MachinePrecision] + N[(NaChar / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -9.5e+138], t$95$2, If[LessEqual[Vef, -4e+58], N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, -2.5e-38], t$95$1, If[LessEqual[Vef, -1.85e-136], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 8e+74], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -9.49999999999999998e138 or 7.99999999999999961e74 < 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 89.0%

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

   6. Taylor expanded in Vef around inf 78.0%

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

   if -9.49999999999999998e138 < Vef < -3.99999999999999978e58

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Vef around inf 69.7%

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

   6. Taylor expanded in Vef around 0 75.8%

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

      1. +-commutative 75.8%

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

   8. Simplified 75.8%

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

   if -3.99999999999999978e58 < Vef < -2.50000000000000017e-38 or -1.85e-136 < Vef < 7.99999999999999961e74

   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 60.5%

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

      1. +-commutative 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in mu around -inf 67.8%

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

      1. associate-*r* 67.8%

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

      2. mul-1-neg 67.8%

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

      3. +-commutative 67.8%

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

      4. mul-1-neg 67.8%

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

      5. unsub-neg 67.8%

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

   10. Simplified 67.8%

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

   if -2.50000000000000017e-38 < Vef < -1.85e-136

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 94.4%

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

   6. Taylor expanded in Ev around inf 68.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -9.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq -4 \cdot 10^{+58}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{elif}\;Vef \leq -1.85 \cdot 10^{-136}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Vef \leq 8 \cdot 10^{+74}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{if}\;mu \leq -2.4 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq -7 \cdot 10^{-42}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(\left(1 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{elif}\;mu \leq 2.9 \cdot 10^{+94}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{Ec}{-KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ mu (- KbT))))))))
   (if (<= mu -2.4e+109)
     t_0
     (if (<= mu -7e-42)
       (+
        (/ NaChar (+ 1.0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT))))
        (/
         NdChar
         (+
          1.0
          (-
           (+ 1.0 (+ (/ EDonor KbT) (+ (/ Vef KbT) (/ mu KbT))))
           (/ Ec KbT)))))
       (if (<= mu 2e-52)
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (/
           NaChar
           (-
            1.0
            (*
             mu
             (+
              (/ 1.0 KbT)
              (/
               (- -1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
               mu))))))
         (if (<= mu 2.9e+94)
           (-
            (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
            (/ NdChar (- -1.0 (exp (/ Ec (- KbT))))))
           t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	double tmp;
	if (mu <= -2.4e+109) {
		tmp = t_0;
	} else if (mu <= -7e-42) {
		tmp = (NaChar / (1.0 + exp(((Vef - ((mu - EAccept) - Ev)) / KbT)))) + (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT))));
	} else if (mu <= 2e-52) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else if (mu <= 2.9e+94) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - (NdChar / (-1.0 - exp((Ec / -KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((mu / -kbt))))
    if (mu <= (-2.4d+109)) then
        tmp = t_0
    else if (mu <= (-7d-42)) then
        tmp = (nachar / (1.0d0 + exp(((vef - ((mu - eaccept) - ev)) / kbt)))) + (ndchar / (1.0d0 + ((1.0d0 + ((edonor / kbt) + ((vef / kbt) + (mu / kbt)))) - (ec / kbt))))
    else if (mu <= 2d-52) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / (1.0d0 - (mu * ((1.0d0 / kbt) + (((-1.0d0) - ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) / mu)))))
    else if (mu <= 2.9d+94) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) - (ndchar / ((-1.0d0) - exp((ec / -kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	double tmp;
	if (mu <= -2.4e+109) {
		tmp = t_0;
	} else if (mu <= -7e-42) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT)))) + (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT))));
	} else if (mu <= 2e-52) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else if (mu <= 2.9e+94) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) - (NdChar / (-1.0 - Math.exp((Ec / -KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((mu / -KbT))))
	tmp = 0
	if mu <= -2.4e+109:
		tmp = t_0
	elif mu <= -7e-42:
		tmp = (NaChar / (1.0 + math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT)))) + (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT))))
	elif mu <= 2e-52:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))))
	elif mu <= 2.9e+94:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) - (NdChar / (-1.0 - math.exp((Ec / -KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))))
	tmp = 0.0
	if (mu <= -2.4e+109)
		tmp = t_0;
	elseif (mu <= -7e-42)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EDonor / KbT) + Float64(Float64(Vef / KbT) + Float64(mu / KbT)))) - Float64(Ec / KbT)))));
	elseif (mu <= 2e-52)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 - Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Float64(-1.0 - Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) / mu))))));
	elseif (mu <= 2.9e+94)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Ec / Float64(-KbT))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	tmp = 0.0;
	if (mu <= -2.4e+109)
		tmp = t_0;
	elseif (mu <= -7e-42)
		tmp = (NaChar / (1.0 + exp(((Vef - ((mu - EAccept) - Ev)) / KbT)))) + (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT))));
	elseif (mu <= 2e-52)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	elseif (mu <= 2.9e+94)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - (NdChar / (-1.0 - exp((Ec / -KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(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, -2.4e+109], t$95$0, If[LessEqual[mu, -7e-42], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(N[(1.0 + N[(N[(EDonor / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2e-52], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 - N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(-1.0 - N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2.9e+94], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -2.39999999999999987e109 or 2.8999999999999998e94 < 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 84.1%

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

   6. Taylor expanded in mu around inf 77.8%

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

      1. neg-mul-1 77.8%

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

      2. distribute-neg-frac2 77.8%

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

   8. Simplified 77.8%

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

   if -2.39999999999999987e109 < mu < -7.0000000000000004e-42

   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 69.6%

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

   if -7.0000000000000004e-42 < mu < 2e-52

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 63.4%

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

      1. +-commutative 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in mu around -inf 75.2%

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

      1. associate-*r* 75.2%

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

      2. mul-1-neg 75.2%

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

      3. +-commutative 75.2%

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

      4. mul-1-neg 75.2%

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

      5. unsub-neg 75.2%

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

   10. Simplified 75.2%

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

   if 2e-52 < mu < 2.8999999999999998e94

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 91.1%

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

      1. associate-*r/ 91.1%

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

      2. mul-1-neg 91.1%

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

   7. Simplified 91.1%

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

   8. Taylor expanded in Ev around inf 71.4%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq -7 \cdot 10^{-42}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(\left(1 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{elif}\;mu \leq 2.9 \cdot 10^{+94}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{Ec}{-KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{if}\;mu \leq -2 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq -1.02 \cdot 10^{-41}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(\left(1 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 7 \cdot 10^{+94}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ mu (- KbT))))))))
   (if (<= mu -2e+109)
     t_0
     (if (<= mu -1.02e-41)
       (+
        (/ NaChar (+ 1.0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT))))
        (/
         NdChar
         (+
          1.0
          (-
           (+ 1.0 (+ (/ EDonor KbT) (+ (/ Vef KbT) (/ mu KbT))))
           (/ Ec KbT)))))
       (if (<= mu 7e+94)
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (/
           NaChar
           (-
            1.0
            (*
             mu
             (+
              (/ 1.0 KbT)
              (/
               (- -1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
               mu))))))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	double tmp;
	if (mu <= -2e+109) {
		tmp = t_0;
	} else if (mu <= -1.02e-41) {
		tmp = (NaChar / (1.0 + exp(((Vef - ((mu - EAccept) - Ev)) / KbT)))) + (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT))));
	} else if (mu <= 7e+94) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((mu / -kbt))))
    if (mu <= (-2d+109)) then
        tmp = t_0
    else if (mu <= (-1.02d-41)) then
        tmp = (nachar / (1.0d0 + exp(((vef - ((mu - eaccept) - ev)) / kbt)))) + (ndchar / (1.0d0 + ((1.0d0 + ((edonor / kbt) + ((vef / kbt) + (mu / kbt)))) - (ec / kbt))))
    else if (mu <= 7d+94) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / (1.0d0 - (mu * ((1.0d0 / kbt) + (((-1.0d0) - ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) / mu)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	double tmp;
	if (mu <= -2e+109) {
		tmp = t_0;
	} else if (mu <= -1.02e-41) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT)))) + (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT))));
	} else if (mu <= 7e+94) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((mu / -KbT))))
	tmp = 0
	if mu <= -2e+109:
		tmp = t_0
	elif mu <= -1.02e-41:
		tmp = (NaChar / (1.0 + math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT)))) + (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT))))
	elif mu <= 7e+94:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 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 <= -2e+109)
		tmp = t_0;
	elseif (mu <= -1.02e-41)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EDonor / KbT) + Float64(Float64(Vef / KbT) + Float64(mu / KbT)))) - Float64(Ec / KbT)))));
	elseif (mu <= 7e+94)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 - Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Float64(-1.0 - Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) / mu))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	tmp = 0.0;
	if (mu <= -2e+109)
		tmp = t_0;
	elseif (mu <= -1.02e-41)
		tmp = (NaChar / (1.0 + exp(((Vef - ((mu - EAccept) - Ev)) / KbT)))) + (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT))));
	elseif (mu <= 7e+94)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(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, -2e+109], t$95$0, If[LessEqual[mu, -1.02e-41], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(N[(1.0 + N[(N[(EDonor / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 7e+94], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 - N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(-1.0 - N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -1.99999999999999996e109 or 6.9999999999999994e94 < 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 84.1%

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

   6. Taylor expanded in mu around inf 77.8%

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

      1. neg-mul-1 77.8%

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

      2. distribute-neg-frac2 77.8%

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

   8. Simplified 77.8%

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

   if -1.99999999999999996e109 < mu < -1.02e-41

   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 69.6%

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

   if -1.02e-41 < mu < 6.9999999999999994e94

   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 63.4%

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

      1. +-commutative 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in mu around -inf 72.3%

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

      1. associate-*r* 72.3%

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

      2. mul-1-neg 72.3%

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

      3. +-commutative 72.3%

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

      4. mul-1-neg 72.3%

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

      5. unsub-neg 72.3%

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

   10. Simplified 72.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2 \cdot 10^{+109}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq -1.02 \cdot 10^{-41}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(\left(1 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 7 \cdot 10^{+94}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

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(((Vef - ((mu - EAccept) - Ev)) / KbT)))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / 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 = (nachar / (1.0d0 + exp(((vef - ((mu - eaccept) - ev)) / kbt)))) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / 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 (NaChar / (1.0 + Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT)))) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 10: 62.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}\\ \mathbf{if}\;NdChar \leq -3.7:\\ \;\;\;\;\frac{NaChar}{\left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) + 2} - t\_0\\ \mathbf{elif}\;NdChar \leq 3.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - t\_1}\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept - Ev \cdot \left(-1 - \frac{Vef}{Ev}\right)\right) - mu} - t\_0\\ \mathbf{elif}\;NdChar \leq 6.4 \cdot 10^{-37} \lor \neg \left(NdChar \leq 2.85 \cdot 10^{+80}\right) \land NdChar \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;\frac{NaChar}{1 + t\_1} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(2 - Ev \cdot \left(\frac{-1}{KbT} - \frac{Vef}{KbT \cdot Ev}\right)\right) - \frac{mu}{KbT}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
        (t_1 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT))))
   (if (<= NdChar -3.7)
     (- (/ NaChar (+ (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))) 2.0)) t_0)
     (if (<= NdChar 3.2e-193)
       (- (/ NdChar (+ (/ Vef KbT) 2.0)) (/ NaChar (- -1.0 t_1)))
       (if (<= NdChar 2.5e-178)
         (-
          (/ (* KbT NaChar) (- (- EAccept (* Ev (- -1.0 (/ Vef Ev)))) mu))
          t_0)
         (if (or (<= NdChar 6.4e-37)
                 (and (not (<= NdChar 2.85e+80)) (<= NdChar 1.28e+154)))
           (+ (/ NaChar (+ 1.0 t_1)) (/ NdChar (+ 1.0 (- 1.0 (/ Ec KbT)))))
           (-
            (/
             NaChar
             (- (- 2.0 (* Ev (- (/ -1.0 KbT) (/ Vef (* KbT Ev))))) (/ mu KbT)))
            t_0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double tmp;
	if (NdChar <= -3.7) {
		tmp = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0;
	} else if (NdChar <= 3.2e-193) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_1));
	} else if (NdChar <= 2.5e-178) {
		tmp = ((KbT * NaChar) / ((EAccept - (Ev * (-1.0 - (Vef / Ev)))) - mu)) - t_0;
	} else if ((NdChar <= 6.4e-37) || (!(NdChar <= 2.85e+80) && (NdChar <= 1.28e+154))) {
		tmp = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else {
		tmp = (NaChar / ((2.0 - (Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev))))) - (mu / KbT))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt)))
    t_1 = exp(((vef - ((mu - eaccept) - ev)) / kbt))
    if (ndchar <= (-3.7d0)) then
        tmp = (nachar / (((eaccept / kbt) + ((vef / kbt) + (ev / kbt))) + 2.0d0)) - t_0
    else if (ndchar <= 3.2d-193) then
        tmp = (ndchar / ((vef / kbt) + 2.0d0)) - (nachar / ((-1.0d0) - t_1))
    else if (ndchar <= 2.5d-178) then
        tmp = ((kbt * nachar) / ((eaccept - (ev * ((-1.0d0) - (vef / ev)))) - mu)) - t_0
    else if ((ndchar <= 6.4d-37) .or. (.not. (ndchar <= 2.85d+80)) .and. (ndchar <= 1.28d+154)) then
        tmp = (nachar / (1.0d0 + t_1)) + (ndchar / (1.0d0 + (1.0d0 - (ec / kbt))))
    else
        tmp = (nachar / ((2.0d0 - (ev * (((-1.0d0) / kbt) - (vef / (kbt * ev))))) - (mu / kbt))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double tmp;
	if (NdChar <= -3.7) {
		tmp = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0;
	} else if (NdChar <= 3.2e-193) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_1));
	} else if (NdChar <= 2.5e-178) {
		tmp = ((KbT * NaChar) / ((EAccept - (Ev * (-1.0 - (Vef / Ev)))) - mu)) - t_0;
	} else if ((NdChar <= 6.4e-37) || (!(NdChar <= 2.85e+80) && (NdChar <= 1.28e+154))) {
		tmp = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else {
		tmp = (NaChar / ((2.0 - (Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev))))) - (mu / KbT))) - t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))
	t_1 = math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))
	tmp = 0
	if NdChar <= -3.7:
		tmp = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0
	elif NdChar <= 3.2e-193:
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_1))
	elif NdChar <= 2.5e-178:
		tmp = ((KbT * NaChar) / ((EAccept - (Ev * (-1.0 - (Vef / Ev)))) - mu)) - t_0
	elif (NdChar <= 6.4e-37) or (not (NdChar <= 2.85e+80) and (NdChar <= 1.28e+154)):
		tmp = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))))
	else:
		tmp = (NaChar / ((2.0 - (Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev))))) - (mu / KbT))) - t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	t_1 = exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT))
	tmp = 0.0
	if (NdChar <= -3.7)
		tmp = Float64(Float64(NaChar / Float64(Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) + 2.0)) - t_0);
	elseif (NdChar <= 3.2e-193)
		tmp = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 2.0)) - Float64(NaChar / Float64(-1.0 - t_1)));
	elseif (NdChar <= 2.5e-178)
		tmp = Float64(Float64(Float64(KbT * NaChar) / Float64(Float64(EAccept - Float64(Ev * Float64(-1.0 - Float64(Vef / Ev)))) - mu)) - t_0);
	elseif ((NdChar <= 6.4e-37) || (!(NdChar <= 2.85e+80) && (NdChar <= 1.28e+154)))
		tmp = Float64(Float64(NaChar / Float64(1.0 + t_1)) + Float64(NdChar / Float64(1.0 + Float64(1.0 - Float64(Ec / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(Float64(2.0 - Float64(Ev * Float64(Float64(-1.0 / KbT) - Float64(Vef / Float64(KbT * Ev))))) - Float64(mu / KbT))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	t_1 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	tmp = 0.0;
	if (NdChar <= -3.7)
		tmp = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0;
	elseif (NdChar <= 3.2e-193)
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_1));
	elseif (NdChar <= 2.5e-178)
		tmp = ((KbT * NaChar) / ((EAccept - (Ev * (-1.0 - (Vef / Ev)))) - mu)) - t_0;
	elseif ((NdChar <= 6.4e-37) || (~((NdChar <= 2.85e+80)) && (NdChar <= 1.28e+154)))
		tmp = (NaChar / (1.0 + t_1)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	else
		tmp = (NaChar / ((2.0 - (Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev))))) - (mu / KbT))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[NdChar, -3.7], N[(N[(NaChar / N[(N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[NdChar, 3.2e-193], N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.5e-178], N[(N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(EAccept - N[(Ev * N[(-1.0 - N[(Vef / Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[Or[LessEqual[NdChar, 6.4e-37], And[N[Not[LessEqual[NdChar, 2.85e+80]], $MachinePrecision], LessEqual[NdChar, 1.28e+154]]], N[(N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(N[(2.0 - N[(Ev * N[(N[(-1.0 / KbT), $MachinePrecision] - N[(Vef / N[(KbT * Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -3.7000000000000002

   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 66.5%

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

      1. +-commutative 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in mu around 0 71.8%

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

   if -3.7000000000000002 < NdChar < 3.20000000000000006e-193

   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 83.5%

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

   6. Taylor expanded in Vef around 0 68.8%

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

      1. +-commutative 68.8%

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

   8. Simplified 68.8%

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

   if 3.20000000000000006e-193 < NdChar < 2.49999999999999988e-178

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in KbT around inf 64.2%

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

      1. +-commutative 64.2%

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

   7. Simplified 64.2%

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

   8. Taylor expanded in Ev around inf 64.2%

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

   9. Taylor expanded in KbT around 0 75.8%

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

   if 2.49999999999999988e-178 < NdChar < 6.3999999999999998e-37 or 2.84999999999999985e80 < NdChar < 1.2800000000000001e154

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 68.8%

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

      1. associate-*r/ 68.8%

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

      2. mul-1-neg 68.8%

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

   7. Simplified 68.8%

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

   8. Taylor expanded in Ec around 0 67.5%

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

      1. neg-mul-1 67.5%

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

      2. unsub-neg 67.5%

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

   10. Simplified 67.5%

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

   if 6.3999999999999998e-37 < NdChar < 2.84999999999999985e80 or 1.2800000000000001e154 < 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 64.9%

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

      1. +-commutative 64.9%

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

   7. Simplified 64.9%

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

   8. Taylor expanded in Ev around inf 70.8%

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

   9. Taylor expanded in EAccept around 0 73.0%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -3.7:\\ \;\;\;\;\frac{NaChar}{\left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) + 2} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 3.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept - Ev \cdot \left(-1 - \frac{Vef}{Ev}\right)\right) - mu} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 6.4 \cdot 10^{-37} \lor \neg \left(NdChar \leq 2.85 \cdot 10^{+80}\right) \land NdChar \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(2 - Ev \cdot \left(\frac{-1}{KbT} - \frac{Vef}{KbT \cdot Ev}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}\\ t_1 := \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{if}\;EAccept \leq -7.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;EAccept \leq 1.1 \cdot 10^{-39}:\\ \;\;\;\;\frac{NaChar}{1 - \left(\frac{mu}{KbT} - \left(1 - \left(Vef \cdot \left(\frac{-1}{KbT} - \frac{Ev}{Vef \cdot KbT}\right) - \frac{EAccept}{KbT}\right)\right)\right)} - t\_1\\ \mathbf{elif}\;EAccept \leq 2.7 \cdot 10^{+149} \lor \neg \left(EAccept \leq 5 \cdot 10^{+245}\right):\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 - \frac{Ev \cdot \left(-1 - \frac{Vef}{Ev}\right) - EAccept}{KbT}\right) - \frac{mu}{KbT}\right)} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT)))
        (t_1 (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))))
   (if (<= EAccept -7.6e-206)
     (+ (/ NaChar (+ 1.0 t_0)) (/ NdChar (+ 1.0 (- 1.0 (/ Ec KbT)))))
     (if (<= EAccept 1.1e-39)
       (-
        (/
         NaChar
         (-
          1.0
          (-
           (/ mu KbT)
           (-
            1.0
            (- (* Vef (- (/ -1.0 KbT) (/ Ev (* Vef KbT)))) (/ EAccept KbT))))))
        t_1)
       (if (or (<= EAccept 2.7e+149) (not (<= EAccept 5e+245)))
         (- (/ NdChar (+ (/ Vef KbT) 2.0)) (/ NaChar (- -1.0 t_0)))
         (-
          (/
           NaChar
           (+
            1.0
            (-
             (- 1.0 (/ (- (* Ev (- -1.0 (/ Vef Ev))) EAccept) KbT))
             (/ mu 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 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_1 = NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double tmp;
	if (EAccept <= -7.6e-206) {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else if (EAccept <= 1.1e-39) {
		tmp = (NaChar / (1.0 - ((mu / KbT) - (1.0 - ((Vef * ((-1.0 / KbT) - (Ev / (Vef * KbT)))) - (EAccept / KbT)))))) - t_1;
	} else if ((EAccept <= 2.7e+149) || !(EAccept <= 5e+245)) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((vef - ((mu - eaccept) - ev)) / kbt))
    t_1 = ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt)))
    if (eaccept <= (-7.6d-206)) then
        tmp = (nachar / (1.0d0 + t_0)) + (ndchar / (1.0d0 + (1.0d0 - (ec / kbt))))
    else if (eaccept <= 1.1d-39) then
        tmp = (nachar / (1.0d0 - ((mu / kbt) - (1.0d0 - ((vef * (((-1.0d0) / kbt) - (ev / (vef * kbt)))) - (eaccept / kbt)))))) - t_1
    else if ((eaccept <= 2.7d+149) .or. (.not. (eaccept <= 5d+245))) then
        tmp = (ndchar / ((vef / kbt) + 2.0d0)) - (nachar / ((-1.0d0) - t_0))
    else
        tmp = (nachar / (1.0d0 + ((1.0d0 - (((ev * ((-1.0d0) - (vef / ev))) - eaccept) / kbt)) - (mu / kbt)))) - t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_1 = NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double tmp;
	if (EAccept <= -7.6e-206) {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else if (EAccept <= 1.1e-39) {
		tmp = (NaChar / (1.0 - ((mu / KbT) - (1.0 - ((Vef * ((-1.0 / KbT) - (Ev / (Vef * KbT)))) - (EAccept / KbT)))))) - t_1;
	} else if ((EAccept <= 2.7e+149) || !(EAccept <= 5e+245)) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))
	t_1 = NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))
	tmp = 0
	if EAccept <= -7.6e-206:
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))))
	elif EAccept <= 1.1e-39:
		tmp = (NaChar / (1.0 - ((mu / KbT) - (1.0 - ((Vef * ((-1.0 / KbT) - (Ev / (Vef * KbT)))) - (EAccept / KbT)))))) - t_1
	elif (EAccept <= 2.7e+149) or not (EAccept <= 5e+245):
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_0))
	else:
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT))
	t_1 = Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	tmp = 0.0
	if (EAccept <= -7.6e-206)
		tmp = Float64(Float64(NaChar / Float64(1.0 + t_0)) + Float64(NdChar / Float64(1.0 + Float64(1.0 - Float64(Ec / KbT)))));
	elseif (EAccept <= 1.1e-39)
		tmp = Float64(Float64(NaChar / Float64(1.0 - Float64(Float64(mu / KbT) - Float64(1.0 - Float64(Float64(Vef * Float64(Float64(-1.0 / KbT) - Float64(Ev / Float64(Vef * KbT)))) - Float64(EAccept / KbT)))))) - t_1);
	elseif ((EAccept <= 2.7e+149) || !(EAccept <= 5e+245))
		tmp = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 2.0)) - Float64(NaChar / Float64(-1.0 - t_0)));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 - Float64(Float64(Float64(Ev * Float64(-1.0 - Float64(Vef / Ev))) - EAccept) / KbT)) - Float64(mu / KbT)))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	t_1 = NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	tmp = 0.0;
	if (EAccept <= -7.6e-206)
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	elseif (EAccept <= 1.1e-39)
		tmp = (NaChar / (1.0 - ((mu / KbT) - (1.0 - ((Vef * ((-1.0 / KbT) - (Ev / (Vef * KbT)))) - (EAccept / KbT)))))) - t_1;
	elseif ((EAccept <= 2.7e+149) || ~((EAccept <= 5e+245)))
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_0));
	else
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EAccept, -7.6e-206], N[(N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EAccept, 1.1e-39], N[(N[(NaChar / N[(1.0 - N[(N[(mu / KbT), $MachinePrecision] - N[(1.0 - N[(N[(Vef * N[(N[(-1.0 / KbT), $MachinePrecision] - N[(Ev / N[(Vef * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[Or[LessEqual[EAccept, 2.7e+149], N[Not[LessEqual[EAccept, 5e+245]], $MachinePrecision]], N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[(N[(1.0 - N[(N[(N[(Ev * N[(-1.0 - N[(Vef / Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if EAccept < -7.60000000000000005e-206

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 75.5%

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

      1. associate-*r/ 75.5%

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

      2. mul-1-neg 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in Ec around 0 55.6%

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

      1. neg-mul-1 55.6%

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

      2. unsub-neg 55.6%

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

   10. Simplified 55.6%

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

   if -7.60000000000000005e-206 < EAccept < 1.1e-39

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 62.0%

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

      1. +-commutative 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in Vef around inf 67.4%

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

   if 1.1e-39 < EAccept < 2.7000000000000001e149 or 5.00000000000000034e245 < EAccept

   1. Initial program 100.0%

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

   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 75.8%

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

   6. Taylor expanded in Vef around 0 60.7%

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

      1. +-commutative 60.7%

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

   8. Simplified 60.7%

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

   if 2.7000000000000001e149 < EAccept < 5.00000000000000034e245

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.4%

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

      1. +-commutative 71.4%

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

   7. Simplified 71.4%

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

   8. Taylor expanded in Ev around inf 65.5%

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

   9. Taylor expanded in KbT around 0 77.1%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq -7.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;EAccept \leq 1.1 \cdot 10^{-39}:\\ \;\;\;\;\frac{NaChar}{1 - \left(\frac{mu}{KbT} - \left(1 - \left(Vef \cdot \left(\frac{-1}{KbT} - \frac{Ev}{Vef \cdot KbT}\right) - \frac{EAccept}{KbT}\right)\right)\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;EAccept \leq 2.7 \cdot 10^{+149} \lor \neg \left(EAccept \leq 5 \cdot 10^{+245}\right):\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 - \frac{Ev \cdot \left(-1 - \frac{Vef}{Ev}\right) - EAccept}{KbT}\right) - \frac{mu}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}\\ \mathbf{if}\;EAccept \leq -9 \cdot 10^{-206}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;EAccept \leq 1.16 \cdot 10^{-38} \lor \neg \left(EAccept \leq 10^{+149}\right) \land EAccept \leq 4.6 \cdot 10^{+245}:\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 - \frac{Ev \cdot \left(-1 - \frac{Vef}{Ev}\right) - EAccept}{KbT}\right) - \frac{mu}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT))))
   (if (<= EAccept -9e-206)
     (+ (/ NaChar (+ 1.0 t_0)) (/ NdChar (+ 1.0 (- 1.0 (/ Ec KbT)))))
     (if (or (<= EAccept 1.16e-38)
             (and (not (<= EAccept 1e+149)) (<= EAccept 4.6e+245)))
       (-
        (/
         NaChar
         (+
          1.0
          (-
           (- 1.0 (/ (- (* Ev (- -1.0 (/ Vef Ev))) EAccept) KbT))
           (/ mu KbT))))
        (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
       (- (/ NdChar (+ (/ Vef KbT) 2.0)) (/ NaChar (- -1.0 t_0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double tmp;
	if (EAccept <= -9e-206) {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else if ((EAccept <= 1.16e-38) || (!(EAccept <= 1e+149) && (EAccept <= 4.6e+245))) {
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	} else {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((vef - ((mu - eaccept) - ev)) / kbt))
    if (eaccept <= (-9d-206)) then
        tmp = (nachar / (1.0d0 + t_0)) + (ndchar / (1.0d0 + (1.0d0 - (ec / kbt))))
    else if ((eaccept <= 1.16d-38) .or. (.not. (eaccept <= 1d+149)) .and. (eaccept <= 4.6d+245)) then
        tmp = (nachar / (1.0d0 + ((1.0d0 - (((ev * ((-1.0d0) - (vef / ev))) - eaccept) / kbt)) - (mu / kbt)))) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt))))
    else
        tmp = (ndchar / ((vef / kbt) + 2.0d0)) - (nachar / ((-1.0d0) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double tmp;
	if (EAccept <= -9e-206) {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else if ((EAccept <= 1.16e-38) || (!(EAccept <= 1e+149) && (EAccept <= 4.6e+245))) {
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	} else {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))
	tmp = 0
	if EAccept <= -9e-206:
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))))
	elif (EAccept <= 1.16e-38) or (not (EAccept <= 1e+149) and (EAccept <= 4.6e+245)):
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	else:
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT))
	tmp = 0.0
	if (EAccept <= -9e-206)
		tmp = Float64(Float64(NaChar / Float64(1.0 + t_0)) + Float64(NdChar / Float64(1.0 + Float64(1.0 - Float64(Ec / KbT)))));
	elseif ((EAccept <= 1.16e-38) || (!(EAccept <= 1e+149) && (EAccept <= 4.6e+245)))
		tmp = Float64(Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 - Float64(Float64(Float64(Ev * Float64(-1.0 - Float64(Vef / Ev))) - EAccept) / KbT)) - Float64(mu / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))));
	else
		tmp = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 2.0)) - Float64(NaChar / Float64(-1.0 - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	tmp = 0.0;
	if (EAccept <= -9e-206)
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	elseif ((EAccept <= 1.16e-38) || (~((EAccept <= 1e+149)) && (EAccept <= 4.6e+245)))
		tmp = (NaChar / (1.0 + ((1.0 - (((Ev * (-1.0 - (Vef / Ev))) - EAccept) / KbT)) - (mu / KbT)))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	else
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[EAccept, -9e-206], N[(N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[EAccept, 1.16e-38], And[N[Not[LessEqual[EAccept, 1e+149]], $MachinePrecision], LessEqual[EAccept, 4.6e+245]]], N[(N[(NaChar / N[(1.0 + N[(N[(1.0 - N[(N[(N[(Ev * N[(-1.0 - N[(Vef / Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if EAccept < -8.9999999999999996e-206

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 75.5%

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

      1. associate-*r/ 75.5%

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

      2. mul-1-neg 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in Ec around 0 55.6%

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

      1. neg-mul-1 55.6%

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

      2. unsub-neg 55.6%

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

   10. Simplified 55.6%

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

   if -8.9999999999999996e-206 < EAccept < 1.15999999999999995e-38 or 1.00000000000000005e149 < EAccept < 4.5999999999999999e245

   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 63.0%

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

      1. +-commutative 63.0%

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

   7. Simplified 63.0%

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

   8. Taylor expanded in Ev around inf 62.3%

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

   9. Taylor expanded in KbT around 0 68.6%

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

   if 1.15999999999999995e-38 < EAccept < 1.00000000000000005e149 or 4.5999999999999999e245 < EAccept

   1. Initial program 100.0%

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

   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 75.3%

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

   6. Taylor expanded in Vef around 0 59.8%

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

      1. +-commutative 59.8%

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

   8. Simplified 59.8%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq -9 \cdot 10^{-206}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;EAccept \leq 1.16 \cdot 10^{-38} \lor \neg \left(EAccept \leq 10^{+149}\right) \land EAccept \leq 4.6 \cdot 10^{+245}:\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 - \frac{Ev \cdot \left(-1 - \frac{Vef}{Ev}\right) - EAccept}{KbT}\right) - \frac{mu}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := \frac{NaChar}{\left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) + 2} - t\_0\\ t_2 := e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}\\ t_3 := \frac{NaChar}{1 + t\_2} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{if}\;NdChar \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 3.4 \cdot 10^{-193}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - t\_2}\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept - Ev \cdot \left(-1 - \frac{Vef}{Ev}\right)\right) - mu} - t\_0\\ \mathbf{elif}\;NdChar \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;NdChar \leq 1.35 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 3 \cdot 10^{+154}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
        (t_1
         (-
          (/ NaChar (+ (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))) 2.0))
          t_0))
        (t_2 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT)))
        (t_3 (+ (/ NaChar (+ 1.0 t_2)) (/ NdChar (+ 1.0 (- 1.0 (/ Ec KbT)))))))
   (if (<= NdChar -2.9e-8)
     t_1
     (if (<= NdChar 3.4e-193)
       (- (/ NdChar (+ (/ Vef KbT) 2.0)) (/ NaChar (- -1.0 t_2)))
       (if (<= NdChar 2.5e-178)
         (-
          (/ (* KbT NaChar) (- (- EAccept (* Ev (- -1.0 (/ Vef Ev)))) mu))
          t_0)
         (if (<= NdChar 1.3e-36)
           t_3
           (if (<= NdChar 1.35e+98)
             t_1
             (if (<= NdChar 3e+154) t_3 (- (* NaChar 0.5) t_0)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0;
	double t_2 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_3 = (NaChar / (1.0 + t_2)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	double tmp;
	if (NdChar <= -2.9e-8) {
		tmp = t_1;
	} else if (NdChar <= 3.4e-193) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_2));
	} else if (NdChar <= 2.5e-178) {
		tmp = ((KbT * NaChar) / ((EAccept - (Ev * (-1.0 - (Vef / Ev)))) - mu)) - t_0;
	} else if (NdChar <= 1.3e-36) {
		tmp = t_3;
	} else if (NdChar <= 1.35e+98) {
		tmp = t_1;
	} else if (NdChar <= 3e+154) {
		tmp = t_3;
	} else {
		tmp = (NaChar * 0.5) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt)))
    t_1 = (nachar / (((eaccept / kbt) + ((vef / kbt) + (ev / kbt))) + 2.0d0)) - t_0
    t_2 = exp(((vef - ((mu - eaccept) - ev)) / kbt))
    t_3 = (nachar / (1.0d0 + t_2)) + (ndchar / (1.0d0 + (1.0d0 - (ec / kbt))))
    if (ndchar <= (-2.9d-8)) then
        tmp = t_1
    else if (ndchar <= 3.4d-193) then
        tmp = (ndchar / ((vef / kbt) + 2.0d0)) - (nachar / ((-1.0d0) - t_2))
    else if (ndchar <= 2.5d-178) then
        tmp = ((kbt * nachar) / ((eaccept - (ev * ((-1.0d0) - (vef / ev)))) - mu)) - t_0
    else if (ndchar <= 1.3d-36) then
        tmp = t_3
    else if (ndchar <= 1.35d+98) then
        tmp = t_1
    else if (ndchar <= 3d+154) then
        tmp = t_3
    else
        tmp = (nachar * 0.5d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0;
	double t_2 = Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double t_3 = (NaChar / (1.0 + t_2)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	double tmp;
	if (NdChar <= -2.9e-8) {
		tmp = t_1;
	} else if (NdChar <= 3.4e-193) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_2));
	} else if (NdChar <= 2.5e-178) {
		tmp = ((KbT * NaChar) / ((EAccept - (Ev * (-1.0 - (Vef / Ev)))) - mu)) - t_0;
	} else if (NdChar <= 1.3e-36) {
		tmp = t_3;
	} else if (NdChar <= 1.35e+98) {
		tmp = t_1;
	} else if (NdChar <= 3e+154) {
		tmp = t_3;
	} else {
		tmp = (NaChar * 0.5) - t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))
	t_1 = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0
	t_2 = math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))
	t_3 = (NaChar / (1.0 + t_2)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))))
	tmp = 0
	if NdChar <= -2.9e-8:
		tmp = t_1
	elif NdChar <= 3.4e-193:
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_2))
	elif NdChar <= 2.5e-178:
		tmp = ((KbT * NaChar) / ((EAccept - (Ev * (-1.0 - (Vef / Ev)))) - mu)) - t_0
	elif NdChar <= 1.3e-36:
		tmp = t_3
	elif NdChar <= 1.35e+98:
		tmp = t_1
	elif NdChar <= 3e+154:
		tmp = t_3
	else:
		tmp = (NaChar * 0.5) - t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	t_1 = Float64(Float64(NaChar / Float64(Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) + 2.0)) - t_0)
	t_2 = exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT))
	t_3 = Float64(Float64(NaChar / Float64(1.0 + t_2)) + Float64(NdChar / Float64(1.0 + Float64(1.0 - Float64(Ec / KbT)))))
	tmp = 0.0
	if (NdChar <= -2.9e-8)
		tmp = t_1;
	elseif (NdChar <= 3.4e-193)
		tmp = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 2.0)) - Float64(NaChar / Float64(-1.0 - t_2)));
	elseif (NdChar <= 2.5e-178)
		tmp = Float64(Float64(Float64(KbT * NaChar) / Float64(Float64(EAccept - Float64(Ev * Float64(-1.0 - Float64(Vef / Ev)))) - mu)) - t_0);
	elseif (NdChar <= 1.3e-36)
		tmp = t_3;
	elseif (NdChar <= 1.35e+98)
		tmp = t_1;
	elseif (NdChar <= 3e+154)
		tmp = t_3;
	else
		tmp = Float64(Float64(NaChar * 0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	t_1 = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0;
	t_2 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	t_3 = (NaChar / (1.0 + t_2)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	tmp = 0.0;
	if (NdChar <= -2.9e-8)
		tmp = t_1;
	elseif (NdChar <= 3.4e-193)
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_2));
	elseif (NdChar <= 2.5e-178)
		tmp = ((KbT * NaChar) / ((EAccept - (Ev * (-1.0 - (Vef / Ev)))) - mu)) - t_0;
	elseif (NdChar <= 1.3e-36)
		tmp = t_3;
	elseif (NdChar <= 1.35e+98)
		tmp = t_1;
	elseif (NdChar <= 3e+154)
		tmp = t_3;
	else
		tmp = (NaChar * 0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(NaChar / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -2.9e-8], t$95$1, If[LessEqual[NdChar, 3.4e-193], N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.5e-178], N[(N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(EAccept - N[(Ev * N[(-1.0 - N[(Vef / Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[NdChar, 1.3e-36], t$95$3, If[LessEqual[NdChar, 1.35e+98], t$95$1, If[LessEqual[NdChar, 3e+154], t$95$3, N[(N[(NaChar * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -2.9000000000000002e-8 or 1.3e-36 < NdChar < 1.35e98

   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 66.8%

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

      1. +-commutative 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in mu around 0 70.9%

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

   if -2.9000000000000002e-8 < NdChar < 3.4000000000000002e-193

   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 83.5%

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

   6. Taylor expanded in Vef around 0 68.8%

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

      1. +-commutative 68.8%

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

   8. Simplified 68.8%

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

   if 3.4000000000000002e-193 < NdChar < 2.49999999999999988e-178

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in KbT around inf 64.2%

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

      1. +-commutative 64.2%

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

   7. Simplified 64.2%

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

   8. Taylor expanded in Ev around inf 64.2%

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

   9. Taylor expanded in KbT around 0 75.8%

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

   if 2.49999999999999988e-178 < NdChar < 1.3e-36 or 1.35e98 < NdChar < 3.00000000000000026e154

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 73.5%

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

      1. associate-*r/ 73.5%

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

      2. mul-1-neg 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in Ec around 0 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   10. Simplified 69.7%

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

   if 3.00000000000000026e154 < 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 58.8%

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

      1. +-commutative 58.8%

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

   7. Simplified 58.8%

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

   8. Taylor expanded in Ev around inf 62.6%

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

   9. Taylor expanded in KbT around inf 63.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \color{blue}{0.5 \cdot NaChar} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{NaChar}{\left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) + 2} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 3.4 \cdot 10^{-193}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept - Ev \cdot \left(-1 - \frac{Vef}{Ev}\right)\right) - mu} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 1.35 \cdot 10^{+98}:\\ \;\;\;\;\frac{NaChar}{\left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) + 2} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 3 \cdot 10^{+154}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}\\ \mathbf{if}\;mu \leq -6.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 1.36 \cdot 10^{+218}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 - \left(-1 - \frac{EDonor}{KbT}\right)} - \frac{NaChar}{-1 - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT))))
   (if (<= mu -6.2e+88)
     (+ (/ NaChar (+ 1.0 t_0)) (/ NdChar (+ 1.0 (- 1.0 (/ Ec KbT)))))
     (if (<= mu 1.36e+218)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
        (/
         NaChar
         (-
          1.0
          (*
           mu
           (+
            (/ 1.0 KbT)
            (/ (- -1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT)))) mu))))))
       (-
        (/ NdChar (- 1.0 (- -1.0 (/ EDonor KbT))))
        (/ NaChar (- -1.0 t_0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double tmp;
	if (mu <= -6.2e+88) {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else if (mu <= 1.36e+218) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else {
		tmp = (NdChar / (1.0 - (-1.0 - (EDonor / KbT)))) - (NaChar / (-1.0 - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((vef - ((mu - eaccept) - ev)) / kbt))
    if (mu <= (-6.2d+88)) then
        tmp = (nachar / (1.0d0 + t_0)) + (ndchar / (1.0d0 + (1.0d0 - (ec / kbt))))
    else if (mu <= 1.36d+218) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / (1.0d0 - (mu * ((1.0d0 / kbt) + (((-1.0d0) - ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) / mu)))))
    else
        tmp = (ndchar / (1.0d0 - ((-1.0d0) - (edonor / kbt)))) - (nachar / ((-1.0d0) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double tmp;
	if (mu <= -6.2e+88) {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else if (mu <= 1.36e+218) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	} else {
		tmp = (NdChar / (1.0 - (-1.0 - (EDonor / KbT)))) - (NaChar / (-1.0 - t_0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))
	tmp = 0
	if mu <= -6.2e+88:
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))))
	elif mu <= 1.36e+218:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))))
	else:
		tmp = (NdChar / (1.0 - (-1.0 - (EDonor / KbT)))) - (NaChar / (-1.0 - t_0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT))
	tmp = 0.0
	if (mu <= -6.2e+88)
		tmp = Float64(Float64(NaChar / Float64(1.0 + t_0)) + Float64(NdChar / Float64(1.0 + Float64(1.0 - Float64(Ec / KbT)))));
	elseif (mu <= 1.36e+218)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 - Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Float64(-1.0 - Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) / mu))))));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 - Float64(-1.0 - Float64(EDonor / KbT)))) - Float64(NaChar / Float64(-1.0 - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	tmp = 0.0;
	if (mu <= -6.2e+88)
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	elseif (mu <= 1.36e+218)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 - (mu * ((1.0 / KbT) + ((-1.0 - ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu)))));
	else
		tmp = (NdChar / (1.0 - (-1.0 - (EDonor / KbT)))) - (NaChar / (-1.0 - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[mu, -6.2e+88], N[(N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.36e+218], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 - N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(-1.0 - N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 - N[(-1.0 - N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -6.2000000000000003e88

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Ec around inf 70.7%

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

      1. associate-*r/ 70.7%

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

      2. mul-1-neg 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in Ec around 0 59.6%

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

      1. neg-mul-1 59.6%

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

      2. unsub-neg 59.6%

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

   10. Simplified 59.6%

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

   if -6.2000000000000003e88 < mu < 1.36e218

   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 62.3%

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

      1. +-commutative 62.3%

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

   7. Simplified 62.3%

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

   8. Taylor expanded in mu around -inf 68.8%

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

      1. associate-*r* 68.8%

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

      2. mul-1-neg 68.8%

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

      3. +-commutative 68.8%

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

      4. mul-1-neg 68.8%

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

      5. unsub-neg 68.8%

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

   10. Simplified 68.8%

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

   if 1.36e218 < 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 EDonor around inf 57.2%

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

   6. Taylor expanded in EDonor around 0 58.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -6.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 1.36 \cdot 10^{+218}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 - mu \cdot \left(\frac{1}{KbT} + \frac{-1 - \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 - \left(-1 - \frac{EDonor}{KbT}\right)} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := \frac{NaChar}{\left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) + 2} - t\_0\\ t_2 := e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}\\ \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - t\_2}\\ \mathbf{elif}\;NdChar \leq 2.6 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 8.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{NaChar}{1 + t\_2} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
        (t_1
         (-
          (/ NaChar (+ (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))) 2.0))
          t_0))
        (t_2 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT))))
   (if (<= NdChar -2.4e-6)
     t_1
     (if (<= NdChar 2.5e-36)
       (- (/ NdChar (+ (/ Vef KbT) 2.0)) (/ NaChar (- -1.0 t_2)))
       (if (<= NdChar 2.6e+98)
         t_1
         (if (<= NdChar 8.8e+154)
           (+ (/ NaChar (+ 1.0 t_2)) (/ NdChar (+ 1.0 (- 1.0 (/ Ec KbT)))))
           (- (* NaChar 0.5) t_0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0;
	double t_2 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double tmp;
	if (NdChar <= -2.4e-6) {
		tmp = t_1;
	} else if (NdChar <= 2.5e-36) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_2));
	} else if (NdChar <= 2.6e+98) {
		tmp = t_1;
	} else if (NdChar <= 8.8e+154) {
		tmp = (NaChar / (1.0 + t_2)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else {
		tmp = (NaChar * 0.5) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt)))
    t_1 = (nachar / (((eaccept / kbt) + ((vef / kbt) + (ev / kbt))) + 2.0d0)) - t_0
    t_2 = exp(((vef - ((mu - eaccept) - ev)) / kbt))
    if (ndchar <= (-2.4d-6)) then
        tmp = t_1
    else if (ndchar <= 2.5d-36) then
        tmp = (ndchar / ((vef / kbt) + 2.0d0)) - (nachar / ((-1.0d0) - t_2))
    else if (ndchar <= 2.6d+98) then
        tmp = t_1
    else if (ndchar <= 8.8d+154) then
        tmp = (nachar / (1.0d0 + t_2)) + (ndchar / (1.0d0 + (1.0d0 - (ec / kbt))))
    else
        tmp = (nachar * 0.5d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0;
	double t_2 = Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	double tmp;
	if (NdChar <= -2.4e-6) {
		tmp = t_1;
	} else if (NdChar <= 2.5e-36) {
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_2));
	} else if (NdChar <= 2.6e+98) {
		tmp = t_1;
	} else if (NdChar <= 8.8e+154) {
		tmp = (NaChar / (1.0 + t_2)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	} else {
		tmp = (NaChar * 0.5) - t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))
	t_1 = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0
	t_2 = math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))
	tmp = 0
	if NdChar <= -2.4e-6:
		tmp = t_1
	elif NdChar <= 2.5e-36:
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_2))
	elif NdChar <= 2.6e+98:
		tmp = t_1
	elif NdChar <= 8.8e+154:
		tmp = (NaChar / (1.0 + t_2)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))))
	else:
		tmp = (NaChar * 0.5) - t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	t_1 = Float64(Float64(NaChar / Float64(Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) + 2.0)) - t_0)
	t_2 = exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT))
	tmp = 0.0
	if (NdChar <= -2.4e-6)
		tmp = t_1;
	elseif (NdChar <= 2.5e-36)
		tmp = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 2.0)) - Float64(NaChar / Float64(-1.0 - t_2)));
	elseif (NdChar <= 2.6e+98)
		tmp = t_1;
	elseif (NdChar <= 8.8e+154)
		tmp = Float64(Float64(NaChar / Float64(1.0 + t_2)) + Float64(NdChar / Float64(1.0 + Float64(1.0 - Float64(Ec / KbT)))));
	else
		tmp = Float64(Float64(NaChar * 0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	t_1 = (NaChar / (((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT))) + 2.0)) - t_0;
	t_2 = exp(((Vef - ((mu - EAccept) - Ev)) / KbT));
	tmp = 0.0;
	if (NdChar <= -2.4e-6)
		tmp = t_1;
	elseif (NdChar <= 2.5e-36)
		tmp = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - t_2));
	elseif (NdChar <= 2.6e+98)
		tmp = t_1;
	elseif (NdChar <= 8.8e+154)
		tmp = (NaChar / (1.0 + t_2)) + (NdChar / (1.0 + (1.0 - (Ec / KbT))));
	else
		tmp = (NaChar * 0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[NdChar, -2.4e-6], t$95$1, If[LessEqual[NdChar, 2.5e-36], N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.6e+98], t$95$1, If[LessEqual[NdChar, 8.8e+154], N[(N[(NaChar / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -2.3999999999999999e-6 or 2.50000000000000002e-36 < NdChar < 2.6e98

   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 66.8%

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

      1. +-commutative 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in mu around 0 70.9%

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

   if -2.3999999999999999e-6 < NdChar < 2.50000000000000002e-36

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 76.0%

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

   6. Taylor expanded in Vef around 0 64.0%

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

      1. +-commutative 64.0%

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

   8. Simplified 64.0%

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

   if 2.6e98 < NdChar < 8.8000000000000004e154

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 68.3%

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

      1. associate-*r/ 68.3%

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

      2. mul-1-neg 68.3%

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

   7. Simplified 68.3%

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

   8. Taylor expanded in Ec around 0 76.5%

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

      1. neg-mul-1 76.5%

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

      2. unsub-neg 76.5%

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

   10. Simplified 76.5%

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

   if 8.8000000000000004e154 < 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 58.8%

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

      1. +-commutative 58.8%

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

   7. Simplified 58.8%

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

   8. Taylor expanded in Ev around inf 62.6%

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

   9. Taylor expanded in KbT around inf 63.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{NaChar}{\left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) + 2} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 2.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{NaChar}{\left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) + 2} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 8.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 - \frac{Ec}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ t_1 := e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}\\ t_2 := NaChar \cdot 0.5 - \frac{NdChar}{-1 - t\_1}\\ \mathbf{if}\;NdChar \leq -1.16 \cdot 10^{-16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NdChar \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 40000000000000:\\ \;\;\;\;\frac{NdChar}{1 + t\_1} + \left(\left(1 - KbT \cdot \frac{NaChar}{mu}\right) + -1\right)\\ \mathbf{elif}\;NdChar \leq 2.7 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NdChar (+ (/ Vef KbT) 2.0))
          (/ NaChar (- -1.0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT))))))
        (t_1 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))
        (t_2 (- (* NaChar 0.5) (/ NdChar (- -1.0 t_1)))))
   (if (<= NdChar -1.16e-16)
     t_2
     (if (<= NdChar 1.2e-36)
       t_0
       (if (<= NdChar 40000000000000.0)
         (+ (/ NdChar (+ 1.0 t_1)) (+ (- 1.0 (* KbT (/ NaChar mu))) -1.0))
         (if (<= NdChar 2.7e+40) t_0 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	double t_1 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_2 = (NaChar * 0.5) - (NdChar / (-1.0 - t_1));
	double tmp;
	if (NdChar <= -1.16e-16) {
		tmp = t_2;
	} else if (NdChar <= 1.2e-36) {
		tmp = t_0;
	} else if (NdChar <= 40000000000000.0) {
		tmp = (NdChar / (1.0 + t_1)) + ((1.0 - (KbT * (NaChar / mu))) + -1.0);
	} else if (NdChar <= 2.7e+40) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (ndchar / ((vef / kbt) + 2.0d0)) - (nachar / ((-1.0d0) - exp(((vef - ((mu - eaccept) - ev)) / kbt))))
    t_1 = exp(((edonor + (mu + (vef - ec))) / kbt))
    t_2 = (nachar * 0.5d0) - (ndchar / ((-1.0d0) - t_1))
    if (ndchar <= (-1.16d-16)) then
        tmp = t_2
    else if (ndchar <= 1.2d-36) then
        tmp = t_0
    else if (ndchar <= 40000000000000.0d0) then
        tmp = (ndchar / (1.0d0 + t_1)) + ((1.0d0 - (kbt * (nachar / mu))) + (-1.0d0))
    else if (ndchar <= 2.7d+40) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	double t_1 = Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_2 = (NaChar * 0.5) - (NdChar / (-1.0 - t_1));
	double tmp;
	if (NdChar <= -1.16e-16) {
		tmp = t_2;
	} else if (NdChar <= 1.2e-36) {
		tmp = t_0;
	} else if (NdChar <= 40000000000000.0) {
		tmp = (NdChar / (1.0 + t_1)) + ((1.0 - (KbT * (NaChar / mu))) + -1.0);
	} else if (NdChar <= 2.7e+40) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))))
	t_1 = math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))
	t_2 = (NaChar * 0.5) - (NdChar / (-1.0 - t_1))
	tmp = 0
	if NdChar <= -1.16e-16:
		tmp = t_2
	elif NdChar <= 1.2e-36:
		tmp = t_0
	elif NdChar <= 40000000000000.0:
		tmp = (NdChar / (1.0 + t_1)) + ((1.0 - (KbT * (NaChar / mu))) + -1.0)
	elif NdChar <= 2.7e+40:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 2.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT)))))
	t_1 = exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))
	t_2 = Float64(Float64(NaChar * 0.5) - Float64(NdChar / Float64(-1.0 - t_1)))
	tmp = 0.0
	if (NdChar <= -1.16e-16)
		tmp = t_2;
	elseif (NdChar <= 1.2e-36)
		tmp = t_0;
	elseif (NdChar <= 40000000000000.0)
		tmp = Float64(Float64(NdChar / Float64(1.0 + t_1)) + Float64(Float64(1.0 - Float64(KbT * Float64(NaChar / mu))) + -1.0));
	elseif (NdChar <= 2.7e+40)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	t_1 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	t_2 = (NaChar * 0.5) - (NdChar / (-1.0 - t_1));
	tmp = 0.0;
	if (NdChar <= -1.16e-16)
		tmp = t_2;
	elseif (NdChar <= 1.2e-36)
		tmp = t_0;
	elseif (NdChar <= 40000000000000.0)
		tmp = (NdChar / (1.0 + t_1)) + ((1.0 - (KbT * (NaChar / mu))) + -1.0);
	elseif (NdChar <= 2.7e+40)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar * 0.5), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.16e-16], t$95$2, If[LessEqual[NdChar, 1.2e-36], t$95$0, If[LessEqual[NdChar, 40000000000000.0], N[(N[(NdChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[(KbT * N[(NaChar / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.7e+40], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.1600000000000001e-16 or 2.70000000000000009e40 < 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 62.7%

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

      1. +-commutative 62.7%

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

   7. Simplified 62.7%

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

   8. Taylor expanded in Ev around inf 60.4%

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

   9. Taylor expanded in KbT around inf 61.3%

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

   if -1.1600000000000001e-16 < NdChar < 1.2e-36 or 4e13 < NdChar < 2.70000000000000009e40

   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 76.4%

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

   6. Taylor expanded in Vef around 0 64.5%

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

      1. +-commutative 64.5%

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

   8. Simplified 64.5%

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

   if 1.2e-36 < NdChar < 4e13

   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 64.4%

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

      1. +-commutative 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in mu around inf 47.1%

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

      1. associate-*r/ 47.1%

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

      2. mul-1-neg 47.1%

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

      3. distribute-rgt-neg-in 47.1%

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

   10. Simplified 47.1%

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

       1. expm1-log1p-u 46.7%

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

       2. expm1-undefine 72.7%

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

       3. associate-/l* 72.7%

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

   12. Applied egg-rr 72.7%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \color{blue}{\left(e^{\mathsf{log1p}\left(KbT \cdot \frac{-NaChar}{mu}\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. sub-neg 72.7%

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

       2. log1p-undefine 72.7%

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

       3. rem-exp-log 73.0%

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

       4. associate-*r/ 73.1%

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

       5. distribute-rgt-neg-out 73.1%

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

       6. distribute-frac-neg 73.1%

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

       7. unsub-neg 73.1%

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

       8. associate-/l* 73.0%

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

       9. metadata-eval 73.0%

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

   14. Simplified 73.0%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.16 \cdot 10^{-16}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 40000000000000:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \left(\left(1 - KbT \cdot \frac{NaChar}{mu}\right) + -1\right)\\ \mathbf{elif}\;NdChar \leq 2.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := NaChar \cdot 0.5 - t\_0\\ t_2 := \frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -1.1 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 5.3 \cdot 10^{-36}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NdChar \leq 62000:\\ \;\;\;\;\frac{KbT \cdot NaChar}{EAccept} - t\_0\\ \mathbf{elif}\;NdChar \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
        (t_1 (- (* NaChar 0.5) t_0))
        (t_2
         (-
          (/ NdChar (+ (/ Vef KbT) 2.0))
          (/ NaChar (- -1.0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT)))))))
   (if (<= NdChar -1.1e-15)
     t_1
     (if (<= NdChar 5.3e-36)
       t_2
       (if (<= NdChar 62000.0)
         (- (/ (* KbT NaChar) EAccept) t_0)
         (if (<= NdChar 7.5e+41) t_2 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 + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar * 0.5) - t_0;
	double t_2 = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	double tmp;
	if (NdChar <= -1.1e-15) {
		tmp = t_1;
	} else if (NdChar <= 5.3e-36) {
		tmp = t_2;
	} else if (NdChar <= 62000.0) {
		tmp = ((KbT * NaChar) / EAccept) - t_0;
	} else if (NdChar <= 7.5e+41) {
		tmp = t_2;
	} 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 + (mu + (vef - ec))) / kbt)))
    t_1 = (nachar * 0.5d0) - t_0
    t_2 = (ndchar / ((vef / kbt) + 2.0d0)) - (nachar / ((-1.0d0) - exp(((vef - ((mu - eaccept) - ev)) / kbt))))
    if (ndchar <= (-1.1d-15)) then
        tmp = t_1
    else if (ndchar <= 5.3d-36) then
        tmp = t_2
    else if (ndchar <= 62000.0d0) then
        tmp = ((kbt * nachar) / eaccept) - t_0
    else if (ndchar <= 7.5d+41) then
        tmp = t_2
    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 + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar * 0.5) - t_0;
	double t_2 = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	double tmp;
	if (NdChar <= -1.1e-15) {
		tmp = t_1;
	} else if (NdChar <= 5.3e-36) {
		tmp = t_2;
	} else if (NdChar <= 62000.0) {
		tmp = ((KbT * NaChar) / EAccept) - t_0;
	} else if (NdChar <= 7.5e+41) {
		tmp = t_2;
	} 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 + (mu + (Vef - Ec))) / KbT)))
	t_1 = (NaChar * 0.5) - t_0
	t_2 = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))))
	tmp = 0
	if NdChar <= -1.1e-15:
		tmp = t_1
	elif NdChar <= 5.3e-36:
		tmp = t_2
	elif NdChar <= 62000.0:
		tmp = ((KbT * NaChar) / EAccept) - t_0
	elif NdChar <= 7.5e+41:
		tmp = t_2
	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(mu + Float64(Vef - Ec))) / KbT))))
	t_1 = Float64(Float64(NaChar * 0.5) - t_0)
	t_2 = Float64(Float64(NdChar / Float64(Float64(Vef / KbT) + 2.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -1.1e-15)
		tmp = t_1;
	elseif (NdChar <= 5.3e-36)
		tmp = t_2;
	elseif (NdChar <= 62000.0)
		tmp = Float64(Float64(Float64(KbT * NaChar) / EAccept) - t_0);
	elseif (NdChar <= 7.5e+41)
		tmp = t_2;
	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 + (mu + (Vef - Ec))) / KbT)));
	t_1 = (NaChar * 0.5) - t_0;
	t_2 = (NdChar / ((Vef / KbT) + 2.0)) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -1.1e-15)
		tmp = t_1;
	elseif (NdChar <= 5.3e-36)
		tmp = t_2;
	elseif (NdChar <= 62000.0)
		tmp = ((KbT * NaChar) / EAccept) - t_0;
	elseif (NdChar <= 7.5e+41)
		tmp = t_2;
	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[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[(Vef / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.1e-15], t$95$1, If[LessEqual[NdChar, 5.3e-36], t$95$2, If[LessEqual[NdChar, 62000.0], N[(N[(N[(KbT * NaChar), $MachinePrecision] / EAccept), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[NdChar, 7.5e+41], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.09999999999999993e-15 or 7.50000000000000072e41 < 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 62.7%

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

      1. +-commutative 62.7%

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

   7. Simplified 62.7%

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

   8. Taylor expanded in Ev around inf 60.4%

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

   9. Taylor expanded in KbT around inf 61.3%

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

   if -1.09999999999999993e-15 < NdChar < 5.2999999999999998e-36 or 62000 < NdChar < 7.50000000000000072e41

   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 76.4%

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

   6. Taylor expanded in Vef around 0 64.5%

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

      1. +-commutative 64.5%

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

   8. Simplified 64.5%

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

   if 5.2999999999999998e-36 < NdChar < 62000

   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 64.4%

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

      1. +-commutative 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in EAccept around inf 47.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.1 \cdot 10^{-15}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 5.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 62000:\\ \;\;\;\;\frac{KbT \cdot NaChar}{EAccept} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT} + 2} - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -2.9 \cdot 10^{-258}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 2.3 \cdot 10^{-156}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{Vef} - \frac{NdChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.48 \cdot 10^{+19}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (* NdChar 0.5)
          (/ NaChar (- -1.0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT)))))))
   (if (<= NaChar -2.9e-258)
     t_0
     (if (<= NaChar 2.3e-156)
       (- (* KbT (/ NaChar Vef)) (/ NdChar (- -1.0 (exp (/ Vef KbT)))))
       (if (<= NaChar 1.48e+19)
         (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (/ NaChar 2.0))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar * 0.5) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	double tmp;
	if (NaChar <= -2.9e-258) {
		tmp = t_0;
	} else if (NaChar <= 2.3e-156) {
		tmp = (KbT * (NaChar / Vef)) - (NdChar / (-1.0 - exp((Vef / KbT))));
	} else if (NaChar <= 1.48e+19) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar * 0.5d0) - (nachar / ((-1.0d0) - exp(((vef - ((mu - eaccept) - ev)) / kbt))))
    if (nachar <= (-2.9d-258)) then
        tmp = t_0
    else if (nachar <= 2.3d-156) then
        tmp = (kbt * (nachar / vef)) - (ndchar / ((-1.0d0) - exp((vef / kbt))))
    else if (nachar <= 1.48d+19) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	double tmp;
	if (NaChar <= -2.9e-258) {
		tmp = t_0;
	} else if (NaChar <= 2.3e-156) {
		tmp = (KbT * (NaChar / Vef)) - (NdChar / (-1.0 - Math.exp((Vef / KbT))));
	} else if (NaChar <= 1.48e+19) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))))
	tmp = 0
	if NaChar <= -2.9e-258:
		tmp = t_0
	elif NaChar <= 2.3e-156:
		tmp = (KbT * (NaChar / Vef)) - (NdChar / (-1.0 - math.exp((Vef / KbT))))
	elif NaChar <= 1.48e+19:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT)))))
	tmp = 0.0
	if (NaChar <= -2.9e-258)
		tmp = t_0;
	elseif (NaChar <= 2.3e-156)
		tmp = Float64(Float64(KbT * Float64(NaChar / Vef)) - Float64(NdChar / Float64(-1.0 - exp(Float64(Vef / KbT)))));
	elseif (NaChar <= 1.48e+19)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar * 0.5) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	tmp = 0.0;
	if (NaChar <= -2.9e-258)
		tmp = t_0;
	elseif (NaChar <= 2.3e-156)
		tmp = (KbT * (NaChar / Vef)) - (NdChar / (-1.0 - exp((Vef / KbT))));
	elseif (NaChar <= 1.48e+19)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -2.9e-258], t$95$0, If[LessEqual[NaChar, 2.3e-156], N[(N[(KbT * N[(NaChar / Vef), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.48e+19], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -2.9e-258 or 1.48e19 < 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 49.4%

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

   if -2.9e-258 < NaChar < 2.3e-156

   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%

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

      1. +-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in Vef around inf 69.9%

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

      1. associate-/l* 67.3%

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

   10. Simplified 67.3%

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

   11. Taylor expanded in Vef around inf 51.9%

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

   if 2.3e-156 < NaChar < 1.48e19

   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 70.0%

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

   6. Taylor expanded in KbT around inf 57.4%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.9 \cdot 10^{-258}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 2.3 \cdot 10^{-156}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{Vef} - \frac{NdChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.48 \cdot 10^{+19}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -0.155 \lor \neg \left(NaChar \leq 1.26 \cdot 10^{+82}\right):\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -0.155) (not (<= NaChar 1.26e+82)))
   (-
    (* NdChar 0.5)
    (/ NaChar (- -1.0 (exp (/ (- Vef (- (- mu EAccept) Ev)) KbT)))))
   (-
    (* NaChar 0.5)
    (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) 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 ((NaChar <= -0.155) || !(NaChar <= 1.26e+82)) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	} else {
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / 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 ((nachar <= (-0.155d0)) .or. (.not. (nachar <= 1.26d+82))) then
        tmp = (ndchar * 0.5d0) - (nachar / ((-1.0d0) - exp(((vef - ((mu - eaccept) - ev)) / kbt))))
    else
        tmp = (nachar * 0.5d0) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / 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 ((NaChar <= -0.155) || !(NaChar <= 1.26e+82)) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	} else {
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -0.155) or not (NaChar <= 1.26e+82):
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))))
	else:
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -0.155) || !(NaChar <= 1.26e+82))
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT)))));
	else
		tmp = Float64(Float64(NaChar * 0.5) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -0.155) || ~((NaChar <= 1.26e+82)))
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	else
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -0.155], N[Not[LessEqual[NaChar, 1.26e+82]], $MachinePrecision]], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar * 0.5), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -0.154999999999999999 or 1.2600000000000001e82 < 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 51.6%

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

   if -0.154999999999999999 < NaChar < 1.2600000000000001e82

   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 63.6%

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

      1. +-commutative 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in Ev around inf 63.8%

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

   9. Taylor expanded in KbT around inf 59.6%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -0.155 \lor \neg \left(NaChar \leq 1.26 \cdot 10^{+82}\right):\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 56.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -0.3:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - {e}^{\left(\frac{\left(Vef + EAccept\right) + \left(Ev - mu\right)}{KbT}\right)}}\\ \mathbf{elif}\;NaChar \leq 3.2 \cdot 10^{+81}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -0.3)
   (-
    (* NdChar 0.5)
    (/ NaChar (- -1.0 (pow E (/ (+ (+ Vef EAccept) (- Ev mu)) KbT)))))
   (if (<= NaChar 3.2e+81)
     (-
      (* NaChar 0.5)
      (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
     (-
      (* NdChar 0.5)
      (/ NaChar (- -1.0 (exp (/ (- Vef (- (- mu EAccept) 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 (NaChar <= -0.3) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - pow(((double) M_E), (((Vef + EAccept) + (Ev - mu)) / KbT))));
	} else if (NaChar <= 3.2e+81) {
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	}
	return tmp;
}

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -0.3) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.pow(Math.E, (((Vef + EAccept) + (Ev - mu)) / KbT))));
	} else if (NaChar <= 3.2e+81) {
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -0.3:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.pow(math.e, (((Vef + EAccept) + (Ev - mu)) / KbT))))
	elif NaChar <= 3.2e+81:
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	else:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp(((Vef - ((mu - EAccept) - Ev)) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -0.3)
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - (exp(1) ^ Float64(Float64(Float64(Vef + EAccept) + Float64(Ev - mu)) / KbT)))));
	elseif (NaChar <= 3.2e+81)
		tmp = Float64(Float64(NaChar * 0.5) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))));
	else
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef - Float64(Float64(mu - EAccept) - Ev)) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -0.3)
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - (2.71828182845904523536 ^ (((Vef + EAccept) + (Ev - mu)) / KbT))));
	elseif (NaChar <= 3.2e+81)
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	else
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp(((Vef - ((mu - EAccept) - Ev)) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -0.3], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Power[E, N[(N[(N[(Vef + EAccept), $MachinePrecision] + N[(Ev - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 3.2e+81], N[(N[(NaChar * 0.5), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(Vef - N[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -0.299999999999999989

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-1-e 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. unsub-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. associate--l+ 100.0%

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

      7. mul-1-neg 100.0%

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

      8. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in KbT around inf 53.5%

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

   if -0.299999999999999989 < NaChar < 3.2e81

   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 63.6%

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

      1. +-commutative 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in Ev around inf 63.8%

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

   9. Taylor expanded in KbT around inf 59.6%

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

   if 3.2e81 < NaChar

   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 48.7%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -0.3:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - {e}^{\left(\frac{\left(Vef + EAccept\right) + \left(Ev - mu\right)}{KbT}\right)}}\\ \mathbf{elif}\;NaChar \leq 3.2 \cdot 10^{+81}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef - \left(\left(mu - EAccept\right) - Ev\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 38.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -0.042:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;NdChar \leq 4.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{mu}{-KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -0.042)
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (/ NaChar 2.0))
   (if (<= NdChar 4.2e+22)
     (+ (/ NaChar (+ 1.0 (exp (/ mu (- KbT))))) (* NdChar 0.5))
     (+ (/ NdChar (+ 1.0 (exp (/ EDonor 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 (NdChar <= -0.042) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	} else if (NdChar <= 4.2e+22) {
		tmp = (NaChar / (1.0 + exp((mu / -KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + exp((EDonor / 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 (ndchar <= (-0.042d0)) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / 2.0d0)
    else if (ndchar <= 4.2d+22) then
        tmp = (nachar / (1.0d0 + exp((mu / -kbt)))) + (ndchar * 0.5d0)
    else
        tmp = (ndchar / (1.0d0 + exp((edonor / 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 (NdChar <= -0.042) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / 2.0);
	} else if (NdChar <= 4.2e+22) {
		tmp = (NaChar / (1.0 + Math.exp((mu / -KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -0.042:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / 2.0)
	elif NdChar <= 4.2e+22:
		tmp = (NaChar / (1.0 + math.exp((mu / -KbT)))) + (NdChar * 0.5)
	else:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -0.042)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / 2.0));
	elseif (NdChar <= 4.2e+22)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / 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 (NdChar <= -0.042)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	elseif (NdChar <= 4.2e+22)
		tmp = (NaChar / (1.0 + exp((mu / -KbT)))) + (NdChar * 0.5);
	else
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -0.042], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 4.2e+22], N[(N[(NaChar / N[(1.0 + N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -0.0420000000000000026

   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 64.5%

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

   6. Taylor expanded in KbT around inf 49.1%

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

   if -0.0420000000000000026 < NdChar < 4.1999999999999996e22

   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%

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

   6. Taylor expanded in mu around inf 38.9%

      \[\leadsto 0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\color{blue}{-1 \cdot \frac{mu}{KbT}}}} \]
   7. Step-by-step derivation

      1. neg-mul-1 51.0%

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

      2. distribute-neg-frac2 51.0%

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

   8. Simplified 38.9%

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

   if 4.1999999999999996e22 < 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 EDonor around inf 63.1%

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

   6. Taylor expanded in KbT around inf 41.4%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -0.042:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;NdChar \leq 4.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{mu}{-KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 36.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -1.18 \cdot 10^{-222}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Vef \leq 2.4 \cdot 10^{+64}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Vef -1.18e-222)
   (- (* NdChar 0.5) (/ NaChar (- -1.0 (exp (/ Ev KbT)))))
   (if (<= Vef 2.4e+64)
     (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (/ NaChar 2.0))
     (- (* NdChar 0.5) (/ NaChar (- -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 (Vef <= -1.18e-222) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Ev / KbT))));
	} else if (Vef <= 2.4e+64) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-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 (vef <= (-1.18d-222)) then
        tmp = (ndchar * 0.5d0) - (nachar / ((-1.0d0) - exp((ev / kbt))))
    else if (vef <= 2.4d+64) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (ndchar * 0.5d0) - (nachar / ((-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 (Vef <= -1.18e-222) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp((Ev / KbT))));
	} else if (Vef <= 2.4e+64) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp((Vef / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Vef <= -1.18e-222:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp((Ev / KbT))))
	elif Vef <= 2.4e+64:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp((Vef / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Vef <= -1.18e-222)
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - exp(Float64(Ev / KbT)))));
	elseif (Vef <= 2.4e+64)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / 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 (Vef <= -1.18e-222)
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Ev / KbT))));
	elseif (Vef <= 2.4e+64)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Vef / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Vef, -1.18e-222], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 2.4e+64], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -1.18000000000000007e-222

   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 41.9%

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

   6. Taylor expanded in Ev around inf 32.5%

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

   if -1.18000000000000007e-222 < Vef < 2.39999999999999999e64

   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 75.4%

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

   6. Taylor expanded in KbT around inf 46.8%

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

   if 2.39999999999999999e64 < 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 KbT around inf 44.9%

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

   6. Taylor expanded in Vef around inf 38.6%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.18 \cdot 10^{-222}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Vef \leq 2.4 \cdot 10^{+64}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 36.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -5.2 \cdot 10^{-283}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Vef \leq 9.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Vef -5.2e-283)
   (- (* NdChar 0.5) (/ NaChar (- -1.0 (exp (/ Ev KbT)))))
   (if (<= Vef 9.6e+59)
     (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (/ NaChar 2.0))
     (- (* NdChar 0.5) (/ NaChar (- -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 (Vef <= -5.2e-283) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Ev / KbT))));
	} else if (Vef <= 9.6e+59) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-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 (vef <= (-5.2d-283)) then
        tmp = (ndchar * 0.5d0) - (nachar / ((-1.0d0) - exp((ev / kbt))))
    else if (vef <= 9.6d+59) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (ndchar * 0.5d0) - (nachar / ((-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 (Vef <= -5.2e-283) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp((Ev / KbT))));
	} else if (Vef <= 9.6e+59) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp((Vef / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Vef <= -5.2e-283:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp((Ev / KbT))))
	elif Vef <= 9.6e+59:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp((Vef / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Vef <= -5.2e-283)
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - exp(Float64(Ev / KbT)))));
	elseif (Vef <= 9.6e+59)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / 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 (Vef <= -5.2e-283)
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Ev / KbT))));
	elseif (Vef <= 9.6e+59)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Vef / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Vef, -5.2e-283], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 9.6e+59], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -5.2000000000000002e-283

   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 42.3%

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

   6. Taylor expanded in Ev around inf 32.9%

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

   if -5.2000000000000002e-283 < Vef < 9.6000000000000008e59

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 73.0%

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

   6. Taylor expanded in KbT around inf 43.2%

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

   if 9.6000000000000008e59 < 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 KbT around inf 44.9%

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

   6. Taylor expanded in Vef around inf 38.6%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -5.2 \cdot 10^{-283}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Vef \leq 9.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 36.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq 5.1 \cdot 10^{-17}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Vef 5.1e-17)
   (- (* NdChar 0.5) (/ NaChar (- -1.0 (exp (/ Ev KbT)))))
   (- (* NdChar 0.5) (/ NaChar (- -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 (Vef <= 5.1e-17) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Ev / KbT))));
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-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 (vef <= 5.1d-17) then
        tmp = (ndchar * 0.5d0) - (nachar / ((-1.0d0) - exp((ev / kbt))))
    else
        tmp = (ndchar * 0.5d0) - (nachar / ((-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 (Vef <= 5.1e-17) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp((Ev / KbT))));
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp((Vef / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Vef <= 5.1e-17:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp((Ev / KbT))))
	else:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp((Vef / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Vef <= 5.1e-17)
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - exp(Float64(Ev / KbT)))));
	else
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / 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 (Vef <= 5.1e-17)
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Ev / KbT))));
	else
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Vef / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Vef, 5.1e-17], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Vef < 5.1000000000000003e-17

   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 44.3%

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

   6. Taylor expanded in Ev around inf 35.4%

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

   if 5.1000000000000003e-17 < 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 KbT around inf 44.3%

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

   6. Taylor expanded in Vef around inf 37.4%

      \[\leadsto 0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq 5.1 \cdot 10^{-17}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 36.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq 3.05 \cdot 10^{-115}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{EAccept}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept 3.05e-115)
   (- (* NdChar 0.5) (/ NaChar (- -1.0 (exp (/ Ev KbT)))))
   (- (* NdChar 0.5) (/ NaChar (- -1.0 (exp (/ EAccept KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 3.05e-115) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Ev / KbT))));
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((EAccept / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (eaccept <= 3.05d-115) then
        tmp = (ndchar * 0.5d0) - (nachar / ((-1.0d0) - exp((ev / kbt))))
    else
        tmp = (ndchar * 0.5d0) - (nachar / ((-1.0d0) - exp((eaccept / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 3.05e-115) {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp((Ev / KbT))));
	} else {
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - Math.exp((EAccept / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EAccept <= 3.05e-115:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp((Ev / KbT))))
	else:
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - math.exp((EAccept / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EAccept <= 3.05e-115)
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - exp(Float64(Ev / KbT)))));
	else
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - exp(Float64(EAccept / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EAccept <= 3.05e-115)
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((Ev / KbT))));
	else
		tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((EAccept / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EAccept, 3.05e-115], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if EAccept < 3.0499999999999999e-115

   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 42.6%

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

   6. Taylor expanded in Ev around inf 31.1%

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

   if 3.0499999999999999e-115 < EAccept

   1. Initial program 100.0%

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

   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 48.1%

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

   6. Taylor expanded in EAccept around inf 43.3%

      \[\leadsto 0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 3.05 \cdot 10^{-115}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{EAccept}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 35.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{EAccept}{KbT}}} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (- (* NdChar 0.5) (/ NaChar (- -1.0 (exp (/ 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 * 0.5) - (NaChar / (-1.0 - exp((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 * 0.5d0) - (nachar / ((-1.0d0) - exp((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 * 0.5) - (NaChar / (-1.0 - Math.exp((EAccept / KbT))));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar * 0.5) - (NaChar / (-1.0 - math.exp((EAccept / KbT))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(-1.0 - exp(Float64(EAccept / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar * 0.5) - (NaChar / (-1.0 - exp((EAccept / KbT))));
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(EAccept / 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. Taylor expanded in KbT around inf 44.3%

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

6. Taylor expanded in EAccept around inf 34.3%

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

7. Final simplification 34.3%

   \[\leadsto NdChar \cdot 0.5 - \frac{NaChar}{-1 - e^{\frac{EAccept}{KbT}}} \]
8. Add Preprocessing

Alternative 27: 27.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EAccept - \left(\left(mu - Vef\right) - Ev\right)\\ \mathbf{if}\;Ev \leq -1.42 \cdot 10^{-139}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{\frac{-0.5 \cdot \left(t\_0 \cdot \frac{t\_0}{KbT}\right) + \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)}{KbT} - 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(1 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 - \left(\frac{mu}{KbT} + \left(-1 + \left(Ev \cdot \left(\frac{-1}{KbT} - \frac{Vef}{KbT \cdot Ev}\right) - \frac{EAccept}{KbT}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- EAccept (- (- mu Vef) Ev))))
   (if (<= Ev -1.42e-139)
     (-
      (* NdChar 0.5)
      (/
       NaChar
       (-
        (/ (+ (* -0.5 (* t_0 (/ t_0 KbT))) (- (- mu Vef) (+ EAccept Ev))) KbT)
        2.0)))
     (+
      (/
       NdChar
       (+
        1.0
        (- (+ 1.0 (+ (/ EDonor KbT) (+ (/ Vef KbT) (/ mu KbT)))) (/ Ec KbT))))
      (/
       NaChar
       (-
        1.0
        (+
         (/ mu KbT)
         (+
          -1.0
          (-
           (* Ev (- (/ -1.0 KbT) (/ Vef (* KbT Ev))))
           (/ EAccept KbT))))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EAccept - ((mu - Vef) - Ev);
	double tmp;
	if (Ev <= -1.42e-139) {
		tmp = (NdChar * 0.5) - (NaChar / ((((-0.5 * (t_0 * (t_0 / KbT))) + ((mu - Vef) - (EAccept + Ev))) / KbT) - 2.0));
	} else {
		tmp = (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 - ((mu / KbT) + (-1.0 + ((Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev)))) - (EAccept / KbT))))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eaccept - ((mu - vef) - ev)
    if (ev <= (-1.42d-139)) then
        tmp = (ndchar * 0.5d0) - (nachar / (((((-0.5d0) * (t_0 * (t_0 / kbt))) + ((mu - vef) - (eaccept + ev))) / kbt) - 2.0d0))
    else
        tmp = (ndchar / (1.0d0 + ((1.0d0 + ((edonor / kbt) + ((vef / kbt) + (mu / kbt)))) - (ec / kbt)))) + (nachar / (1.0d0 - ((mu / kbt) + ((-1.0d0) + ((ev * (((-1.0d0) / kbt) - (vef / (kbt * ev)))) - (eaccept / kbt))))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EAccept - ((mu - Vef) - Ev);
	double tmp;
	if (Ev <= -1.42e-139) {
		tmp = (NdChar * 0.5) - (NaChar / ((((-0.5 * (t_0 * (t_0 / KbT))) + ((mu - Vef) - (EAccept + Ev))) / KbT) - 2.0));
	} else {
		tmp = (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 - ((mu / KbT) + (-1.0 + ((Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev)))) - (EAccept / KbT))))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = EAccept - ((mu - Vef) - Ev)
	tmp = 0
	if Ev <= -1.42e-139:
		tmp = (NdChar * 0.5) - (NaChar / ((((-0.5 * (t_0 * (t_0 / KbT))) + ((mu - Vef) - (EAccept + Ev))) / KbT) - 2.0))
	else:
		tmp = (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 - ((mu / KbT) + (-1.0 + ((Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev)))) - (EAccept / KbT))))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EAccept - Float64(Float64(mu - Vef) - Ev))
	tmp = 0.0
	if (Ev <= -1.42e-139)
		tmp = Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(Float64(Float64(Float64(-0.5 * Float64(t_0 * Float64(t_0 / KbT))) + Float64(Float64(mu - Vef) - Float64(EAccept + Ev))) / KbT) - 2.0)));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EDonor / KbT) + Float64(Float64(Vef / KbT) + Float64(mu / KbT)))) - Float64(Ec / KbT)))) + Float64(NaChar / Float64(1.0 - Float64(Float64(mu / KbT) + Float64(-1.0 + Float64(Float64(Ev * Float64(Float64(-1.0 / KbT) - Float64(Vef / Float64(KbT * Ev)))) - Float64(EAccept / KbT)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = EAccept - ((mu - Vef) - Ev);
	tmp = 0.0;
	if (Ev <= -1.42e-139)
		tmp = (NdChar * 0.5) - (NaChar / ((((-0.5 * (t_0 * (t_0 / KbT))) + ((mu - Vef) - (EAccept + Ev))) / KbT) - 2.0));
	else
		tmp = (NdChar / (1.0 + ((1.0 + ((EDonor / KbT) + ((Vef / KbT) + (mu / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 - ((mu / KbT) + (-1.0 + ((Ev * ((-1.0 / KbT) - (Vef / (KbT * Ev)))) - (EAccept / KbT))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EAccept - N[(N[(mu - Vef), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ev, -1.42e-139], N[(N[(NdChar * 0.5), $MachinePrecision] - N[(NaChar / N[(N[(N[(N[(-0.5 * N[(t$95$0 * N[(t$95$0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(mu - Vef), $MachinePrecision] - N[(EAccept + Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[(N[(1.0 + N[(N[(EDonor / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 - N[(N[(mu / KbT), $MachinePrecision] + N[(-1.0 + N[(N[(Ev * N[(N[(-1.0 / KbT), $MachinePrecision] - N[(Vef / N[(KbT * Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if Ev < -1.41999999999999997e-139

   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 46.3%

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

   6. Taylor expanded in KbT around -inf 25.5%

      \[\leadsto 0.5 \cdot NdChar + \frac{NaChar}{\color{blue}{2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -0.5 \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 25.5%

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

      2. unsub-neg 25.5%

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

   8. Simplified 25.5%

      \[\leadsto 0.5 \cdot NdChar + \frac{NaChar}{\color{blue}{2 - \frac{-0.5 \cdot \frac{{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}^{2}}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}}} \]
   9. Step-by-step derivation

      1. unpow2 25.5%

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

      2. *-un-lft-identity 25.5%

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

      3. times-frac 29.9%

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

      4. associate-+l+ 29.9%

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

      5. associate-+l+ 29.9%

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

   10. Applied egg-rr 29.9%

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

   if -1.41999999999999997e-139 < 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 55.7%

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

      1. +-commutative 55.7%

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

   7. Simplified 55.7%

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

   8. Taylor expanded in Ev around inf 56.5%

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

   9. Taylor expanded in KbT around inf 30.0%

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

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -1.42 \cdot 10^{-139}:\\ \;\;\;\;NdChar \cdot 0.5 - \frac{NaChar}{\frac{-0.5 \cdot \left(\left(EAccept - \left(\left(mu - Vef\right) - Ev\right)\right) \cdot \frac{EAccept - \left(\left(mu - Vef\right) - Ev\right)}{KbT}\right) + \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)}{KbT} - 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(1 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 - \left(\frac{mu}{KbT} + \left(-1 + \left(Ev \cdot \left(\frac{-1}{KbT} - \frac{Vef}{KbT \cdot Ev}\right) - \frac{EAccept}{KbT}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 27.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EAccept - \left(\left(mu - Vef\right) - Ev\right)\\ NdChar \cdot 0.5 - \frac{NaChar}{\frac{-0.5 \cdot \left(t\_0 \cdot \frac{t\_0}{KbT}\right) + \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)}{KbT} - 2} \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- EAccept (- (- mu Vef) Ev))))
   (-
    (* NdChar 0.5)
    (/
     NaChar
     (-
      (/ (+ (* -0.5 (* t_0 (/ t_0 KbT))) (- (- mu Vef) (+ EAccept Ev))) KbT)
      2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EAccept - ((mu - Vef) - Ev);
	return (NdChar * 0.5) - (NaChar / ((((-0.5 * (t_0 * (t_0 / KbT))) + ((mu - Vef) - (EAccept + Ev))) / KbT) - 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
    real(8) :: t_0
    t_0 = eaccept - ((mu - vef) - ev)
    code = (ndchar * 0.5d0) - (nachar / (((((-0.5d0) * (t_0 * (t_0 / kbt))) + ((mu - vef) - (eaccept + ev))) / kbt) - 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) {
	double t_0 = EAccept - ((mu - Vef) - Ev);
	return (NdChar * 0.5) - (NaChar / ((((-0.5 * (t_0 * (t_0 / KbT))) + ((mu - Vef) - (EAccept + Ev))) / KbT) - 2.0));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = EAccept - ((mu - Vef) - Ev)
	return (NdChar * 0.5) - (NaChar / ((((-0.5 * (t_0 * (t_0 / KbT))) + ((mu - Vef) - (EAccept + Ev))) / KbT) - 2.0))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EAccept - Float64(Float64(mu - Vef) - Ev))
	return Float64(Float64(NdChar * 0.5) - Float64(NaChar / Float64(Float64(Float64(Float64(-0.5 * Float64(t_0 * Float64(t_0 / KbT))) + Float64(Float64(mu - Vef) - Float64(EAccept + Ev))) / KbT) - 2.0)))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = EAccept - ((mu - Vef) - Ev);
	tmp = (NdChar * 0.5) - (NaChar / ((((-0.5 * (t_0 * (t_0 / KbT))) + ((mu - Vef) - (EAccept + Ev))) / KbT) - 2.0));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in KbT around inf 44.3%

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

6. Taylor expanded in KbT around -inf 24.7%

   \[\leadsto 0.5 \cdot NdChar + \frac{NaChar}{\color{blue}{2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -0.5 \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}}} \]
7. Step-by-step derivation

   1. mul-1-neg 24.7%

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

   2. unsub-neg 24.7%

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

8. Simplified 24.7%

   \[\leadsto 0.5 \cdot NdChar + \frac{NaChar}{\color{blue}{2 - \frac{-0.5 \cdot \frac{{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}^{2}}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}}} \]
9. Step-by-step derivation

   1. unpow2 24.7%

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

   2. *-un-lft-identity 24.7%

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

   3. times-frac 28.1%

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

   4. associate-+l+ 28.1%

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

   5. associate-+l+ 28.1%

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

10. Applied egg-rr 28.1%

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

11. Final simplification 28.1%

    \[\leadsto NdChar \cdot 0.5 - \frac{NaChar}{\frac{-0.5 \cdot \left(\left(EAccept - \left(\left(mu - Vef\right) - Ev\right)\right) \cdot \frac{EAccept - \left(\left(mu - Vef\right) - Ev\right)}{KbT}\right) + \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)}{KbT} - 2} \]
12. Add Preprocessing

Alternative 29: 27.2% accurate, 32.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / 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 44.3%

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

6. Taylor expanded in KbT around inf 26.9%

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

7. Final simplification 26.9%

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

Reproduce

herbie shell --seed 2024094 
(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))))))