Bulmash initializePoisson

Percentage Accurate: 99.9% → 99.9%

Time: 22.2s

Alternatives: 31

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: 99.9% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. --lowering--.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 68.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(EAccept + Ev\right) + \left(Vef - mu\right)\\ t_1 := \frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1}\\ t_2 := t\_1 + \frac{NaChar}{2 + \frac{t\_0 + \frac{-0.5 \cdot \left(t\_0 \cdot \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)\right)}{KbT}}{KbT}}\\ t_3 := t\_1 + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;KbT \leq -6.6 \cdot 10^{+106}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;KbT \leq -3.7 \cdot 10^{-196}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;KbT \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{t\_0}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 1.25 \cdot 10^{+199}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ EAccept Ev) (- Vef mu)))
        (t_1 (/ NdChar (+ (exp (/ (+ Vef (+ mu (- EDonor Ec))) KbT)) 1.0)))
        (t_2
         (+
          t_1
          (/
           NaChar
           (+
            2.0
            (/
             (+ t_0 (/ (* -0.5 (* t_0 (- (- mu Vef) (+ EAccept Ev)))) KbT))
             KbT)))))
        (t_3 (+ t_1 (/ NaChar (+ (exp (/ Vef KbT)) 1.0)))))
   (if (<= KbT -6.6e+106)
     t_3
     (if (<= KbT -3.7e-196)
       t_2
       (if (<= KbT 2.9e-19)
         (/ NaChar (+ (exp (/ t_0 KbT)) 1.0))
         (if (<= KbT 1.25e+199) t_2 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 = (EAccept + Ev) + (Vef - mu);
	double t_1 = NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0);
	double t_2 = t_1 + (NaChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((mu - Vef) - (EAccept + Ev)))) / KbT)) / KbT)));
	double t_3 = t_1 + (NaChar / (exp((Vef / KbT)) + 1.0));
	double tmp;
	if (KbT <= -6.6e+106) {
		tmp = t_3;
	} else if (KbT <= -3.7e-196) {
		tmp = t_2;
	} else if (KbT <= 2.9e-19) {
		tmp = NaChar / (exp((t_0 / KbT)) + 1.0);
	} else if (KbT <= 1.25e+199) {
		tmp = t_2;
	} 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 = (eaccept + ev) + (vef - mu)
    t_1 = ndchar / (exp(((vef + (mu + (edonor - ec))) / kbt)) + 1.0d0)
    t_2 = t_1 + (nachar / (2.0d0 + ((t_0 + (((-0.5d0) * (t_0 * ((mu - vef) - (eaccept + ev)))) / kbt)) / kbt)))
    t_3 = t_1 + (nachar / (exp((vef / kbt)) + 1.0d0))
    if (kbt <= (-6.6d+106)) then
        tmp = t_3
    else if (kbt <= (-3.7d-196)) then
        tmp = t_2
    else if (kbt <= 2.9d-19) then
        tmp = nachar / (exp((t_0 / kbt)) + 1.0d0)
    else if (kbt <= 1.25d+199) then
        tmp = t_2
    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 = (EAccept + Ev) + (Vef - mu);
	double t_1 = NdChar / (Math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0);
	double t_2 = t_1 + (NaChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((mu - Vef) - (EAccept + Ev)))) / KbT)) / KbT)));
	double t_3 = t_1 + (NaChar / (Math.exp((Vef / KbT)) + 1.0));
	double tmp;
	if (KbT <= -6.6e+106) {
		tmp = t_3;
	} else if (KbT <= -3.7e-196) {
		tmp = t_2;
	} else if (KbT <= 2.9e-19) {
		tmp = NaChar / (Math.exp((t_0 / KbT)) + 1.0);
	} else if (KbT <= 1.25e+199) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (EAccept + Ev) + (Vef - mu)
	t_1 = NdChar / (math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)
	t_2 = t_1 + (NaChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((mu - Vef) - (EAccept + Ev)))) / KbT)) / KbT)))
	t_3 = t_1 + (NaChar / (math.exp((Vef / KbT)) + 1.0))
	tmp = 0
	if KbT <= -6.6e+106:
		tmp = t_3
	elif KbT <= -3.7e-196:
		tmp = t_2
	elif KbT <= 2.9e-19:
		tmp = NaChar / (math.exp((t_0 / KbT)) + 1.0)
	elif KbT <= 1.25e+199:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(EAccept + Ev) + Float64(Vef - mu))
	t_1 = Float64(NdChar / Float64(exp(Float64(Float64(Vef + Float64(mu + Float64(EDonor - Ec))) / KbT)) + 1.0))
	t_2 = Float64(t_1 + Float64(NaChar / Float64(2.0 + Float64(Float64(t_0 + Float64(Float64(-0.5 * Float64(t_0 * Float64(Float64(mu - Vef) - Float64(EAccept + Ev)))) / KbT)) / KbT))))
	t_3 = Float64(t_1 + Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)))
	tmp = 0.0
	if (KbT <= -6.6e+106)
		tmp = t_3;
	elseif (KbT <= -3.7e-196)
		tmp = t_2;
	elseif (KbT <= 2.9e-19)
		tmp = Float64(NaChar / Float64(exp(Float64(t_0 / KbT)) + 1.0));
	elseif (KbT <= 1.25e+199)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (EAccept + Ev) + (Vef - mu);
	t_1 = NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0);
	t_2 = t_1 + (NaChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((mu - Vef) - (EAccept + Ev)))) / KbT)) / KbT)));
	t_3 = t_1 + (NaChar / (exp((Vef / KbT)) + 1.0));
	tmp = 0.0;
	if (KbT <= -6.6e+106)
		tmp = t_3;
	elseif (KbT <= -3.7e-196)
		tmp = t_2;
	elseif (KbT <= 2.9e-19)
		tmp = NaChar / (exp((t_0 / KbT)) + 1.0);
	elseif (KbT <= 1.25e+199)
		tmp = t_2;
	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[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[Exp[N[(N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar / N[(2.0 + N[(N[(t$95$0 + N[(N[(-0.5 * N[(t$95$0 * N[(N[(mu - Vef), $MachinePrecision] - N[(EAccept + Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -6.6e+106], t$95$3, If[LessEqual[KbT, -3.7e-196], t$95$2, If[LessEqual[KbT, 2.9e-19], N[(NaChar / N[(N[Exp[N[(t$95$0 / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.25e+199], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -6.60000000000000015e106 or 1.25e199 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 92.3%

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

   7. Simplified 92.3%

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

   if -6.60000000000000015e106 < KbT < -3.7000000000000001e-196 or 2.9e-19 < KbT < 1.25e199

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   7. Simplified 74.3%

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

   if -3.7000000000000001e-196 < KbT < 2.9e-19

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6.6 \cdot 10^{+106}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq -3.7 \cdot 10^{-196}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + \frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{elif}\;KbT \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 1.25 \cdot 10^{+199}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + \frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 4: 67.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(EAccept + Ev\right) + \left(Vef - mu\right)\\ t_1 := \frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1}\\ t_2 := t\_1 + \frac{NaChar}{2 + \frac{t\_0 + \frac{-0.5 \cdot \left(t\_0 \cdot \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)\right)}{KbT}}{KbT}}\\ t_3 := t\_1 + \frac{NaChar}{2}\\ \mathbf{if}\;KbT \leq -9 \cdot 10^{+172}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;KbT \leq -8 \cdot 10^{-196}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;KbT \leq 1.65 \cdot 10^{-18}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{t\_0}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 2.95 \cdot 10^{+199}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ EAccept Ev) (- Vef mu)))
        (t_1 (/ NdChar (+ (exp (/ (+ Vef (+ mu (- EDonor Ec))) KbT)) 1.0)))
        (t_2
         (+
          t_1
          (/
           NaChar
           (+
            2.0
            (/
             (+ t_0 (/ (* -0.5 (* t_0 (- (- mu Vef) (+ EAccept Ev)))) KbT))
             KbT)))))
        (t_3 (+ t_1 (/ NaChar 2.0))))
   (if (<= KbT -9e+172)
     t_3
     (if (<= KbT -8e-196)
       t_2
       (if (<= KbT 1.65e-18)
         (/ NaChar (+ (exp (/ t_0 KbT)) 1.0))
         (if (<= KbT 2.95e+199) t_2 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 = (EAccept + Ev) + (Vef - mu);
	double t_1 = NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0);
	double t_2 = t_1 + (NaChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((mu - Vef) - (EAccept + Ev)))) / KbT)) / KbT)));
	double t_3 = t_1 + (NaChar / 2.0);
	double tmp;
	if (KbT <= -9e+172) {
		tmp = t_3;
	} else if (KbT <= -8e-196) {
		tmp = t_2;
	} else if (KbT <= 1.65e-18) {
		tmp = NaChar / (exp((t_0 / KbT)) + 1.0);
	} else if (KbT <= 2.95e+199) {
		tmp = t_2;
	} 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 = (eaccept + ev) + (vef - mu)
    t_1 = ndchar / (exp(((vef + (mu + (edonor - ec))) / kbt)) + 1.0d0)
    t_2 = t_1 + (nachar / (2.0d0 + ((t_0 + (((-0.5d0) * (t_0 * ((mu - vef) - (eaccept + ev)))) / kbt)) / kbt)))
    t_3 = t_1 + (nachar / 2.0d0)
    if (kbt <= (-9d+172)) then
        tmp = t_3
    else if (kbt <= (-8d-196)) then
        tmp = t_2
    else if (kbt <= 1.65d-18) then
        tmp = nachar / (exp((t_0 / kbt)) + 1.0d0)
    else if (kbt <= 2.95d+199) then
        tmp = t_2
    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 = (EAccept + Ev) + (Vef - mu);
	double t_1 = NdChar / (Math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0);
	double t_2 = t_1 + (NaChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((mu - Vef) - (EAccept + Ev)))) / KbT)) / KbT)));
	double t_3 = t_1 + (NaChar / 2.0);
	double tmp;
	if (KbT <= -9e+172) {
		tmp = t_3;
	} else if (KbT <= -8e-196) {
		tmp = t_2;
	} else if (KbT <= 1.65e-18) {
		tmp = NaChar / (Math.exp((t_0 / KbT)) + 1.0);
	} else if (KbT <= 2.95e+199) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (EAccept + Ev) + (Vef - mu)
	t_1 = NdChar / (math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)
	t_2 = t_1 + (NaChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((mu - Vef) - (EAccept + Ev)))) / KbT)) / KbT)))
	t_3 = t_1 + (NaChar / 2.0)
	tmp = 0
	if KbT <= -9e+172:
		tmp = t_3
	elif KbT <= -8e-196:
		tmp = t_2
	elif KbT <= 1.65e-18:
		tmp = NaChar / (math.exp((t_0 / KbT)) + 1.0)
	elif KbT <= 2.95e+199:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(EAccept + Ev) + Float64(Vef - mu))
	t_1 = Float64(NdChar / Float64(exp(Float64(Float64(Vef + Float64(mu + Float64(EDonor - Ec))) / KbT)) + 1.0))
	t_2 = Float64(t_1 + Float64(NaChar / Float64(2.0 + Float64(Float64(t_0 + Float64(Float64(-0.5 * Float64(t_0 * Float64(Float64(mu - Vef) - Float64(EAccept + Ev)))) / KbT)) / KbT))))
	t_3 = Float64(t_1 + Float64(NaChar / 2.0))
	tmp = 0.0
	if (KbT <= -9e+172)
		tmp = t_3;
	elseif (KbT <= -8e-196)
		tmp = t_2;
	elseif (KbT <= 1.65e-18)
		tmp = Float64(NaChar / Float64(exp(Float64(t_0 / KbT)) + 1.0));
	elseif (KbT <= 2.95e+199)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (EAccept + Ev) + (Vef - mu);
	t_1 = NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0);
	t_2 = t_1 + (NaChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((mu - Vef) - (EAccept + Ev)))) / KbT)) / KbT)));
	t_3 = t_1 + (NaChar / 2.0);
	tmp = 0.0;
	if (KbT <= -9e+172)
		tmp = t_3;
	elseif (KbT <= -8e-196)
		tmp = t_2;
	elseif (KbT <= 1.65e-18)
		tmp = NaChar / (exp((t_0 / KbT)) + 1.0);
	elseif (KbT <= 2.95e+199)
		tmp = t_2;
	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[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[Exp[N[(N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar / N[(2.0 + N[(N[(t$95$0 + N[(N[(-0.5 * N[(t$95$0 * N[(N[(mu - Vef), $MachinePrecision] - N[(EAccept + Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -9e+172], t$95$3, If[LessEqual[KbT, -8e-196], t$95$2, If[LessEqual[KbT, 1.65e-18], N[(NaChar / N[(N[Exp[N[(t$95$0 / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.95e+199], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -9.0000000000000004e172 or 2.94999999999999998e199 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf

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

   7. Simplified 89.0%

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

   if -9.0000000000000004e172 < KbT < -8.0000000000000004e-196 or 1.6500000000000001e-18 < KbT < 2.94999999999999998e199

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   7. Simplified 75.2%

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

   if -8.0000000000000004e-196 < KbT < 1.6500000000000001e-18

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -9 \cdot 10^{+172}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq -8 \cdot 10^{-196}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + \frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.65 \cdot 10^{-18}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 2.95 \cdot 10^{+199}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + \frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(mu - Vef\right) - \left(EAccept + Ev\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{if}\;KbT \leq -4.5 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq -8 \cdot 10^{-197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 5.2 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (- (+ mu (+ Vef EDonor)) Ec) KbT)) 1.0)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ Vef (+ mu (- EDonor Ec))) KbT)) 1.0))
          (/ NaChar 2.0))))
   (if (<= KbT -4.5e+148)
     t_1
     (if (<= KbT -8e-197)
       t_0
       (if (<= KbT 3.4e-18)
         (/ NaChar (+ (exp (/ (+ (+ EAccept Ev) (- Vef mu)) KbT)) 1.0))
         (if (<= KbT 5.2e+131) t_0 t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	double t_1 = (NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0);
	double tmp;
	if (KbT <= -4.5e+148) {
		tmp = t_1;
	} else if (KbT <= -8e-197) {
		tmp = t_0;
	} else if (KbT <= 3.4e-18) {
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	} else if (KbT <= 5.2e+131) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (exp((((mu + (vef + edonor)) - ec) / kbt)) + 1.0d0)
    t_1 = (ndchar / (exp(((vef + (mu + (edonor - ec))) / kbt)) + 1.0d0)) + (nachar / 2.0d0)
    if (kbt <= (-4.5d+148)) then
        tmp = t_1
    else if (kbt <= (-8d-197)) then
        tmp = t_0
    else if (kbt <= 3.4d-18) then
        tmp = nachar / (exp((((eaccept + ev) + (vef - mu)) / kbt)) + 1.0d0)
    else if (kbt <= 5.2d+131) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	double t_1 = (NdChar / (Math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0);
	double tmp;
	if (KbT <= -4.5e+148) {
		tmp = t_1;
	} else if (KbT <= -8e-197) {
		tmp = t_0;
	} else if (KbT <= 3.4e-18) {
		tmp = NaChar / (Math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	} else if (KbT <= 5.2e+131) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0)
	t_1 = (NdChar / (math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0)
	tmp = 0
	if KbT <= -4.5e+148:
		tmp = t_1
	elif KbT <= -8e-197:
		tmp = t_0
	elif KbT <= 3.4e-18:
		tmp = NaChar / (math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0)
	elif KbT <= 5.2e+131:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + Float64(Vef + EDonor)) - Ec) / KbT)) + 1.0))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Vef + Float64(mu + Float64(EDonor - Ec))) / KbT)) + 1.0)) + Float64(NaChar / 2.0))
	tmp = 0.0
	if (KbT <= -4.5e+148)
		tmp = t_1;
	elseif (KbT <= -8e-197)
		tmp = t_0;
	elseif (KbT <= 3.4e-18)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) / KbT)) + 1.0));
	elseif (KbT <= 5.2e+131)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	t_1 = (NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)) + (NaChar / 2.0);
	tmp = 0.0;
	if (KbT <= -4.5e+148)
		tmp = t_1;
	elseif (KbT <= -8e-197)
		tmp = t_0;
	elseif (KbT <= 3.4e-18)
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	elseif (KbT <= 5.2e+131)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(N[(mu + N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -4.5e+148], t$95$1, If[LessEqual[KbT, -8e-197], t$95$0, If[LessEqual[KbT, 3.4e-18], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 5.2e+131], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.49999999999999994e148 or 5.2e131 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf

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

   7. Simplified 81.3%

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

   if -4.49999999999999994e148 < KbT < -7.9999999999999999e-197 or 3.40000000000000001e-18 < KbT < 5.2e131

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 73.3%

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

   7. Simplified 73.3%

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

   if -7.9999999999999999e-197 < KbT < 3.40000000000000001e-18

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq -8 \cdot 10^{-197}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 5.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ t_1 := \frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{2}\\ \mathbf{if}\;KbT \leq -3.75 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq -7.2 \cdot 10^{-196}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 8.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 1.25 \cdot 10^{+136}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (- (+ mu (+ Vef EDonor)) Ec) KbT)) 1.0)))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (+ Vef (+ EAccept (- Ev mu))) KbT)) 1.0))
          (/ NdChar 2.0))))
   (if (<= KbT -3.75e+149)
     t_1
     (if (<= KbT -7.2e-196)
       t_0
       (if (<= KbT 8.2e-19)
         (/ NaChar (+ (exp (/ (+ (+ EAccept Ev) (- Vef mu)) KbT)) 1.0))
         (if (<= KbT 1.25e+136) t_0 t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	double t_1 = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / 2.0);
	double tmp;
	if (KbT <= -3.75e+149) {
		tmp = t_1;
	} else if (KbT <= -7.2e-196) {
		tmp = t_0;
	} else if (KbT <= 8.2e-19) {
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	} else if (KbT <= 1.25e+136) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (exp((((mu + (vef + edonor)) - ec) / kbt)) + 1.0d0)
    t_1 = (nachar / (exp(((vef + (eaccept + (ev - mu))) / kbt)) + 1.0d0)) + (ndchar / 2.0d0)
    if (kbt <= (-3.75d+149)) then
        tmp = t_1
    else if (kbt <= (-7.2d-196)) then
        tmp = t_0
    else if (kbt <= 8.2d-19) then
        tmp = nachar / (exp((((eaccept + ev) + (vef - mu)) / kbt)) + 1.0d0)
    else if (kbt <= 1.25d+136) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	double t_1 = (NaChar / (Math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / 2.0);
	double tmp;
	if (KbT <= -3.75e+149) {
		tmp = t_1;
	} else if (KbT <= -7.2e-196) {
		tmp = t_0;
	} else if (KbT <= 8.2e-19) {
		tmp = NaChar / (Math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	} else if (KbT <= 1.25e+136) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0)
	t_1 = (NaChar / (math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / 2.0)
	tmp = 0
	if KbT <= -3.75e+149:
		tmp = t_1
	elif KbT <= -7.2e-196:
		tmp = t_0
	elif KbT <= 8.2e-19:
		tmp = NaChar / (math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0)
	elif KbT <= 1.25e+136:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + Float64(Vef + EDonor)) - Ec) / KbT)) + 1.0))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(EAccept + Float64(Ev - mu))) / KbT)) + 1.0)) + Float64(NdChar / 2.0))
	tmp = 0.0
	if (KbT <= -3.75e+149)
		tmp = t_1;
	elseif (KbT <= -7.2e-196)
		tmp = t_0;
	elseif (KbT <= 8.2e-19)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) / KbT)) + 1.0));
	elseif (KbT <= 1.25e+136)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	t_1 = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / 2.0);
	tmp = 0.0;
	if (KbT <= -3.75e+149)
		tmp = t_1;
	elseif (KbT <= -7.2e-196)
		tmp = t_0;
	elseif (KbT <= 8.2e-19)
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	elseif (KbT <= 1.25e+136)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(N[(mu + N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(Vef + N[(EAccept + N[(Ev - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -3.75e+149], t$95$1, If[LessEqual[KbT, -7.2e-196], t$95$0, If[LessEqual[KbT, 8.2e-19], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.25e+136], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -3.75000000000000016e149 or 1.25e136 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf

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

   7. Simplified 79.1%

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

   if -3.75000000000000016e149 < KbT < -7.2000000000000001e-196 or 8.1999999999999997e-19 < KbT < 1.25e136

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 73.3%

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

   7. Simplified 73.3%

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

   if -7.2000000000000001e-196 < KbT < 8.1999999999999997e-19

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.75 \cdot 10^{+149}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{2}\\ \mathbf{elif}\;KbT \leq -7.2 \cdot 10^{-196}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 8.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 1.25 \cdot 10^{+136}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;mu \leq -5.2 \cdot 10^{+191}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq -3.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{elif}\;mu \leq 4.5 \cdot 10^{-238}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{elif}\;mu \leq 3.15 \cdot 10^{-146}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 2.45 \cdot 10^{+117}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{0 - \frac{mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0)))
   (if (<= mu -5.2e+191)
     (/ NdChar (+ (exp (/ mu KbT)) 1.0))
     (if (<= mu -3.8e-114)
       (/ NaChar t_0)
       (if (<= mu 4.5e-238)
         (/ NdChar t_0)
         (if (<= mu 3.15e-146)
           (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
           (if (<= mu 2.45e+117)
             (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
             (/ NaChar (+ (exp (- 0.0 (/ mu KbT))) 1.0)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((Vef / KbT)) + 1.0;
	double tmp;
	if (mu <= -5.2e+191) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else if (mu <= -3.8e-114) {
		tmp = NaChar / t_0;
	} else if (mu <= 4.5e-238) {
		tmp = NdChar / t_0;
	} else if (mu <= 3.15e-146) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (mu <= 2.45e+117) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((0.0 - (mu / KbT))) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    if (mu <= (-5.2d+191)) then
        tmp = ndchar / (exp((mu / kbt)) + 1.0d0)
    else if (mu <= (-3.8d-114)) then
        tmp = nachar / t_0
    else if (mu <= 4.5d-238) then
        tmp = ndchar / t_0
    else if (mu <= 3.15d-146) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else if (mu <= 2.45d+117) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp((0.0d0 - (mu / kbt))) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp((Vef / KbT)) + 1.0;
	double tmp;
	if (mu <= -5.2e+191) {
		tmp = NdChar / (Math.exp((mu / KbT)) + 1.0);
	} else if (mu <= -3.8e-114) {
		tmp = NaChar / t_0;
	} else if (mu <= 4.5e-238) {
		tmp = NdChar / t_0;
	} else if (mu <= 3.15e-146) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else if (mu <= 2.45e+117) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp((0.0 - (mu / KbT))) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	tmp = 0
	if mu <= -5.2e+191:
		tmp = NdChar / (math.exp((mu / KbT)) + 1.0)
	elif mu <= -3.8e-114:
		tmp = NaChar / t_0
	elif mu <= 4.5e-238:
		tmp = NdChar / t_0
	elif mu <= 3.15e-146:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	elif mu <= 2.45e+117:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp((0.0 - (mu / KbT))) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	tmp = 0.0
	if (mu <= -5.2e+191)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	elseif (mu <= -3.8e-114)
		tmp = Float64(NaChar / t_0);
	elseif (mu <= 4.5e-238)
		tmp = Float64(NdChar / t_0);
	elseif (mu <= 3.15e-146)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (mu <= 2.45e+117)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(0.0 - Float64(mu / KbT))) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp((Vef / KbT)) + 1.0;
	tmp = 0.0;
	if (mu <= -5.2e+191)
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	elseif (mu <= -3.8e-114)
		tmp = NaChar / t_0;
	elseif (mu <= 4.5e-238)
		tmp = NdChar / t_0;
	elseif (mu <= 3.15e-146)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	elseif (mu <= 2.45e+117)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	else
		tmp = NaChar / (exp((0.0 - (mu / KbT))) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[mu, -5.2e+191], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, -3.8e-114], N[(NaChar / t$95$0), $MachinePrecision], If[LessEqual[mu, 4.5e-238], N[(NdChar / t$95$0), $MachinePrecision], If[LessEqual[mu, 3.15e-146], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2.45e+117], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(0.0 - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if mu < -5.2000000000000001e191

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 71.6%

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

   10. Simplified 71.6%

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

   if -5.2000000000000001e191 < mu < -3.7999999999999998e-114

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 49.0%

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

   12. Simplified 49.0%

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

   if -3.7999999999999998e-114 < mu < 4.49999999999999996e-238

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 55.3%

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

   10. Simplified 55.3%

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

   if 4.49999999999999996e-238 < mu < 3.1499999999999999e-146

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 74.8%

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

   9. Simplified 74.8%

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

   10. Taylor expanded in EAccept around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 66.2%

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

   12. Simplified 66.2%

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

   if 3.1499999999999999e-146 < mu < 2.45e117

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 66.0%

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

   7. Simplified 66.0%

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

   8. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 44.5%

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

   10. Simplified 44.5%

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

   if 2.45e117 < 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%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 66.5%

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

   9. Simplified 66.5%

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

   10. Taylor expanded in mu around inf

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

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

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

       2. /-lowering-/.f64 52.1%

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

   12. Simplified 52.1%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{-1 \cdot \frac{mu}{KbT}}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -5.2 \cdot 10^{+191}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq -3.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 4.5 \cdot 10^{-238}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 3.15 \cdot 10^{-146}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 2.45 \cdot 10^{+117}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{0 - \frac{mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{if}\;Ev \leq -1.22 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -9.6 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Ev \leq -9.5 \cdot 10^{-162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Ev \leq -3.6 \cdot 10^{-274}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Ev \leq 3.1 \cdot 10^{-262}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ Vef KbT)) 1.0)))
        (t_1 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))))
   (if (<= Ev -1.22e+75)
     (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
     (if (<= Ev -9.6e-57)
       t_0
       (if (<= Ev -9.5e-162)
         t_1
         (if (<= Ev -3.6e-274)
           t_0
           (if (<= Ev 3.1e-262)
             t_1
             (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((Vef / KbT)) + 1.0);
	double t_1 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double tmp;
	if (Ev <= -1.22e+75) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else if (Ev <= -9.6e-57) {
		tmp = t_0;
	} else if (Ev <= -9.5e-162) {
		tmp = t_1;
	} else if (Ev <= -3.6e-274) {
		tmp = t_0;
	} else if (Ev <= 3.1e-262) {
		tmp = t_1;
	} else {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (exp((vef / kbt)) + 1.0d0)
    t_1 = ndchar / (exp((edonor / kbt)) + 1.0d0)
    if (ev <= (-1.22d+75)) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else if (ev <= (-9.6d-57)) then
        tmp = t_0
    else if (ev <= (-9.5d-162)) then
        tmp = t_1
    else if (ev <= (-3.6d-274)) then
        tmp = t_0
    else if (ev <= 3.1d-262) then
        tmp = t_1
    else
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	double t_1 = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	double tmp;
	if (Ev <= -1.22e+75) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else if (Ev <= -9.6e-57) {
		tmp = t_0;
	} else if (Ev <= -9.5e-162) {
		tmp = t_1;
	} else if (Ev <= -3.6e-274) {
		tmp = t_0;
	} else if (Ev <= 3.1e-262) {
		tmp = t_1;
	} else {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((Vef / KbT)) + 1.0)
	t_1 = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	tmp = 0
	if Ev <= -1.22e+75:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	elif Ev <= -9.6e-57:
		tmp = t_0
	elif Ev <= -9.5e-162:
		tmp = t_1
	elif Ev <= -3.6e-274:
		tmp = t_0
	elif Ev <= 3.1e-262:
		tmp = t_1
	else:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0))
	t_1 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	tmp = 0.0
	if (Ev <= -1.22e+75)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	elseif (Ev <= -9.6e-57)
		tmp = t_0;
	elseif (Ev <= -9.5e-162)
		tmp = t_1;
	elseif (Ev <= -3.6e-274)
		tmp = t_0;
	elseif (Ev <= 3.1e-262)
		tmp = t_1;
	else
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((Vef / KbT)) + 1.0);
	t_1 = NdChar / (exp((EDonor / KbT)) + 1.0);
	tmp = 0.0;
	if (Ev <= -1.22e+75)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	elseif (Ev <= -9.6e-57)
		tmp = t_0;
	elseif (Ev <= -9.5e-162)
		tmp = t_1;
	elseif (Ev <= -3.6e-274)
		tmp = t_0;
	elseif (Ev <= 3.1e-262)
		tmp = t_1;
	else
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ev, -1.22e+75], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, -9.6e-57], t$95$0, If[LessEqual[Ev, -9.5e-162], t$95$1, If[LessEqual[Ev, -3.6e-274], t$95$0, If[LessEqual[Ev, 3.1e-262], t$95$1, N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if Ev < -1.2199999999999999e75

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 58.0%

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

   9. Simplified 58.0%

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

   10. Taylor expanded in Ev around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 49.7%

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

   12. Simplified 49.7%

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

   if -1.2199999999999999e75 < Ev < -9.60000000000000025e-57 or -9.5000000000000004e-162 < Ev < -3.59999999999999983e-274

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 63.9%

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

   9. Simplified 63.9%

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

   10. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 45.8%

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

   12. Simplified 45.8%

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

   if -9.60000000000000025e-57 < Ev < -9.5000000000000004e-162 or -3.59999999999999983e-274 < Ev < 3.0999999999999998e-262

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 44.0%

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

   10. Simplified 44.0%

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

   if 3.0999999999999998e-262 < Ev

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 59.2%

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

   9. Simplified 59.2%

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

   10. Taylor expanded in EAccept around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 39.1%

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

   12. Simplified 39.1%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -1.22 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -9.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -9.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq -3.6 \cdot 10^{-274}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq 3.1 \cdot 10^{-262}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{if}\;NdChar \leq -1.55 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq -0.053:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 4.6 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ (+ EAccept Ev) (- Vef mu)) KbT)) 1.0)))
        (t_1 (/ NdChar (+ (exp (/ (- (+ mu (+ Vef EDonor)) Ec) KbT)) 1.0))))
   (if (<= NdChar -1.55e+130)
     t_0
     (if (<= NdChar -0.053) t_1 (if (<= NdChar 4.6e+74) t_0 t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	double t_1 = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	double tmp;
	if (NdChar <= -1.55e+130) {
		tmp = t_0;
	} else if (NdChar <= -0.053) {
		tmp = t_1;
	} else if (NdChar <= 4.6e+74) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (exp((((eaccept + ev) + (vef - mu)) / kbt)) + 1.0d0)
    t_1 = ndchar / (exp((((mu + (vef + edonor)) - ec) / kbt)) + 1.0d0)
    if (ndchar <= (-1.55d+130)) then
        tmp = t_0
    else if (ndchar <= (-0.053d0)) then
        tmp = t_1
    else if (ndchar <= 4.6d+74) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	double t_1 = NdChar / (Math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	double tmp;
	if (NdChar <= -1.55e+130) {
		tmp = t_0;
	} else if (NdChar <= -0.053) {
		tmp = t_1;
	} else if (NdChar <= 4.6e+74) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0)
	t_1 = NdChar / (math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0)
	tmp = 0
	if NdChar <= -1.55e+130:
		tmp = t_0
	elif NdChar <= -0.053:
		tmp = t_1
	elif NdChar <= 4.6e+74:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) / KbT)) + 1.0))
	t_1 = Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + Float64(Vef + EDonor)) - Ec) / KbT)) + 1.0))
	tmp = 0.0
	if (NdChar <= -1.55e+130)
		tmp = t_0;
	elseif (NdChar <= -0.053)
		tmp = t_1;
	elseif (NdChar <= 4.6e+74)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	t_1 = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	tmp = 0.0;
	if (NdChar <= -1.55e+130)
		tmp = t_0;
	elseif (NdChar <= -0.053)
		tmp = t_1;
	elseif (NdChar <= 4.6e+74)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[Exp[N[(N[(N[(mu + N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.55e+130], t$95$0, If[LessEqual[NdChar, -0.053], t$95$1, If[LessEqual[NdChar, 4.6e+74], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.55e130 or -0.0529999999999999985 < NdChar < 4.5999999999999997e74

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 69.4%

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

   7. Simplified 69.4%

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

   if -1.55e130 < NdChar < -0.0529999999999999985 or 4.5999999999999997e74 < 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%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.55 \cdot 10^{+130}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq -0.053:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 4.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ t_1 := \frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{if}\;mu \leq -2.1 \cdot 10^{+196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;mu \leq -1.25 \cdot 10^{-114}:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{elif}\;mu \leq 3.8 \cdot 10^{-238}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{elif}\;mu \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0))
        (t_1 (/ NdChar (+ (exp (/ mu KbT)) 1.0))))
   (if (<= mu -2.1e+196)
     t_1
     (if (<= mu -1.25e-114)
       (/ NaChar t_0)
       (if (<= mu 3.8e-238)
         (/ NdChar t_0)
         (if (<= mu 3.8e-80) (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)) t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((Vef / KbT)) + 1.0;
	double t_1 = NdChar / (exp((mu / KbT)) + 1.0);
	double tmp;
	if (mu <= -2.1e+196) {
		tmp = t_1;
	} else if (mu <= -1.25e-114) {
		tmp = NaChar / t_0;
	} else if (mu <= 3.8e-238) {
		tmp = NdChar / t_0;
	} else if (mu <= 3.8e-80) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    t_1 = ndchar / (exp((mu / kbt)) + 1.0d0)
    if (mu <= (-2.1d+196)) then
        tmp = t_1
    else if (mu <= (-1.25d-114)) then
        tmp = nachar / t_0
    else if (mu <= 3.8d-238) then
        tmp = ndchar / t_0
    else if (mu <= 3.8d-80) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp((Vef / KbT)) + 1.0;
	double t_1 = NdChar / (Math.exp((mu / KbT)) + 1.0);
	double tmp;
	if (mu <= -2.1e+196) {
		tmp = t_1;
	} else if (mu <= -1.25e-114) {
		tmp = NaChar / t_0;
	} else if (mu <= 3.8e-238) {
		tmp = NdChar / t_0;
	} else if (mu <= 3.8e-80) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	t_1 = NdChar / (math.exp((mu / KbT)) + 1.0)
	tmp = 0
	if mu <= -2.1e+196:
		tmp = t_1
	elif mu <= -1.25e-114:
		tmp = NaChar / t_0
	elif mu <= 3.8e-238:
		tmp = NdChar / t_0
	elif mu <= 3.8e-80:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	t_1 = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0))
	tmp = 0.0
	if (mu <= -2.1e+196)
		tmp = t_1;
	elseif (mu <= -1.25e-114)
		tmp = Float64(NaChar / t_0);
	elseif (mu <= 3.8e-238)
		tmp = Float64(NdChar / t_0);
	elseif (mu <= 3.8e-80)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp((Vef / KbT)) + 1.0;
	t_1 = NdChar / (exp((mu / KbT)) + 1.0);
	tmp = 0.0;
	if (mu <= -2.1e+196)
		tmp = t_1;
	elseif (mu <= -1.25e-114)
		tmp = NaChar / t_0;
	elseif (mu <= 3.8e-238)
		tmp = NdChar / t_0;
	elseif (mu <= 3.8e-80)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -2.1e+196], t$95$1, If[LessEqual[mu, -1.25e-114], N[(NaChar / t$95$0), $MachinePrecision], If[LessEqual[mu, 3.8e-238], N[(NdChar / t$95$0), $MachinePrecision], If[LessEqual[mu, 3.8e-80], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -2.10000000000000015e196 or 3.79999999999999967e-80 < 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%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 62.7%

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

   7. Simplified 62.7%

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

   8. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 52.1%

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

   10. Simplified 52.1%

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

   if -2.10000000000000015e196 < mu < -1.24999999999999997e-114

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 49.0%

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

   12. Simplified 49.0%

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

   if -1.24999999999999997e-114 < mu < 3.7999999999999997e-238

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 55.3%

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

   10. Simplified 55.3%

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

   if 3.7999999999999997e-238 < mu < 3.79999999999999967e-80

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 68.9%

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

   9. Simplified 68.9%

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

   10. Taylor expanded in EAccept around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 60.1%

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

   12. Simplified 60.1%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2.1 \cdot 10^{+196}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq -1.25 \cdot 10^{-114}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 3.8 \cdot 10^{-238}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef + mu\right)\\ t_1 := \frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ t_2 := t\_0 - Ec\\ \mathbf{if}\;KbT \leq -3.6 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq -4.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{t\_2 + -0.5 \cdot \frac{t\_2 \cdot \left(Ec - t\_0\right)}{KbT}}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.02 \cdot 10^{-19}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (+ Vef mu)))
        (t_1
         (+
          (/ NdChar 2.0)
          (/
           NaChar
           (+
            2.0
            (/
             (+
              (+ (+ EAccept Ev) (- Vef mu))
              (*
               -0.5
               (*
                (+ EAccept (+ Ev (- Vef mu)))
                (/ (- (- (- mu Vef) Ev) EAccept) KbT))))
             KbT)))))
        (t_2 (- t_0 Ec)))
   (if (<= KbT -3.6e+144)
     t_1
     (if (<= KbT -4.2e-61)
       (/ NdChar (+ 2.0 (/ (+ t_2 (* -0.5 (/ (* t_2 (- Ec t_0)) KbT))) KbT)))
       (if (<= KbT 1.02e-19) (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)) t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef + mu);
	double t_1 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	double t_2 = t_0 - Ec;
	double tmp;
	if (KbT <= -3.6e+144) {
		tmp = t_1;
	} else if (KbT <= -4.2e-61) {
		tmp = NdChar / (2.0 + ((t_2 + (-0.5 * ((t_2 * (Ec - t_0)) / KbT))) / KbT));
	} else if (KbT <= 1.02e-19) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = edonor + (vef + mu)
    t_1 = (ndchar / 2.0d0) + (nachar / (2.0d0 + ((((eaccept + ev) + (vef - mu)) + ((-0.5d0) * ((eaccept + (ev + (vef - mu))) * ((((mu - vef) - ev) - eaccept) / kbt)))) / kbt)))
    t_2 = t_0 - ec
    if (kbt <= (-3.6d+144)) then
        tmp = t_1
    else if (kbt <= (-4.2d-61)) then
        tmp = ndchar / (2.0d0 + ((t_2 + ((-0.5d0) * ((t_2 * (ec - t_0)) / kbt))) / kbt))
    else if (kbt <= 1.02d-19) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef + mu);
	double t_1 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	double t_2 = t_0 - Ec;
	double tmp;
	if (KbT <= -3.6e+144) {
		tmp = t_1;
	} else if (KbT <= -4.2e-61) {
		tmp = NdChar / (2.0 + ((t_2 + (-0.5 * ((t_2 * (Ec - t_0)) / KbT))) / KbT));
	} else if (KbT <= 1.02e-19) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = EDonor + (Vef + mu)
	t_1 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)))
	t_2 = t_0 - Ec
	tmp = 0
	if KbT <= -3.6e+144:
		tmp = t_1
	elif KbT <= -4.2e-61:
		tmp = NdChar / (2.0 + ((t_2 + (-0.5 * ((t_2 * (Ec - t_0)) / KbT))) / KbT))
	elif KbT <= 1.02e-19:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef + mu))
	t_1 = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) + Float64(-0.5 * Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) * Float64(Float64(Float64(Float64(mu - Vef) - Ev) - EAccept) / KbT)))) / KbT))))
	t_2 = Float64(t_0 - Ec)
	tmp = 0.0
	if (KbT <= -3.6e+144)
		tmp = t_1;
	elseif (KbT <= -4.2e-61)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(t_2 + Float64(-0.5 * Float64(Float64(t_2 * Float64(Ec - t_0)) / KbT))) / KbT)));
	elseif (KbT <= 1.02e-19)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = EDonor + (Vef + mu);
	t_1 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	t_2 = t_0 - Ec;
	tmp = 0.0;
	if (KbT <= -3.6e+144)
		tmp = t_1;
	elseif (KbT <= -4.2e-61)
		tmp = NdChar / (2.0 + ((t_2 + (-0.5 * ((t_2 * (Ec - t_0)) / KbT))) / KbT));
	elseif (KbT <= 1.02e-19)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(mu - Vef), $MachinePrecision] - Ev), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - Ec), $MachinePrecision]}, If[LessEqual[KbT, -3.6e+144], t$95$1, If[LessEqual[KbT, -4.2e-61], N[(NdChar / N[(2.0 + N[(N[(t$95$2 + N[(-0.5 * N[(N[(t$95$2 * N[(Ec - t$95$0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.02e-19], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -3.5999999999999997e144 or 1.02000000000000004e-19 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   7. Simplified 70.9%

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

   8. Taylor expanded in KbT around inf

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

   10. Simplified 40.7%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-/l* N/A

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

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

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

       6. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

       11. associate-+l+ N/A

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

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

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

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

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

       14. --lowering--.f64 55.7%

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

   12. Applied egg-rr 55.7%

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

   if -3.5999999999999997e144 < KbT < -4.1999999999999998e-61

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in KbT around -inf

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

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

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

      2. mul-1-neg N/A

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

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

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

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

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

   10. Simplified 39.0%

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

   if -4.1999999999999998e-61 < KbT < 1.02000000000000004e-19

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 67.4%

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

   9. Simplified 67.4%

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

   10. Taylor expanded in EAccept around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 40.7%

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

   12. Simplified 40.7%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.6 \cdot 10^{+144}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ \mathbf{elif}\;KbT \leq -4.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + -0.5 \cdot \frac{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.02 \cdot 10^{-19}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ \mathbf{if}\;KbT \leq -2.65 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 4.6 \cdot 10^{+196}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar 2.0)
          (/
           NaChar
           (+
            2.0
            (/
             (+
              (+ (+ EAccept Ev) (- Vef mu))
              (*
               -0.5
               (*
                (+ EAccept (+ Ev (- Vef mu)))
                (/ (- (- (- mu Vef) Ev) EAccept) KbT))))
             KbT))))))
   (if (<= KbT -2.65e+122)
     t_0
     (if (<= KbT 4.6e+196)
       (/ NaChar (+ (exp (/ (- (+ Vef Ev) mu) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	double tmp;
	if (KbT <= -2.65e+122) {
		tmp = t_0;
	} else if (KbT <= 4.6e+196) {
		tmp = NaChar / (exp((((Vef + Ev) - mu) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / 2.0d0) + (nachar / (2.0d0 + ((((eaccept + ev) + (vef - mu)) + ((-0.5d0) * ((eaccept + (ev + (vef - mu))) * ((((mu - vef) - ev) - eaccept) / kbt)))) / kbt)))
    if (kbt <= (-2.65d+122)) then
        tmp = t_0
    else if (kbt <= 4.6d+196) then
        tmp = nachar / (exp((((vef + ev) - mu) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	double tmp;
	if (KbT <= -2.65e+122) {
		tmp = t_0;
	} else if (KbT <= 4.6e+196) {
		tmp = NaChar / (Math.exp((((Vef + Ev) - mu) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)))
	tmp = 0
	if KbT <= -2.65e+122:
		tmp = t_0
	elif KbT <= 4.6e+196:
		tmp = NaChar / (math.exp((((Vef + Ev) - mu) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) + Float64(-0.5 * Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) * Float64(Float64(Float64(Float64(mu - Vef) - Ev) - EAccept) / KbT)))) / KbT))))
	tmp = 0.0
	if (KbT <= -2.65e+122)
		tmp = t_0;
	elseif (KbT <= 4.6e+196)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	tmp = 0.0;
	if (KbT <= -2.65e+122)
		tmp = t_0;
	elseif (KbT <= 4.6e+196)
		tmp = NaChar / (exp((((Vef + Ev) - mu) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(mu - Vef), $MachinePrecision] - Ev), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -2.65e+122], t$95$0, If[LessEqual[KbT, 4.6e+196], N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -2.65e122 or 4.59999999999999961e196 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   7. Simplified 67.0%

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

   8. Taylor expanded in KbT around inf

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

   10. Simplified 49.4%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-/l* N/A

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

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       6. associate-+l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       11. associate-+l+ N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       14. --lowering--.f64 73.9%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

   12. Applied egg-rr 73.9%

       \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{2 - \frac{\color{blue}{\left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}\right) \cdot -0.5} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}} \]

   if -2.65e122 < KbT < 4.59999999999999961e196

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 61.5%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in EAccept around 0

       \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}\right)}\right)\right) \]

       3. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(Ev + Vef\right) - mu}{KbT}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(Ev + Vef\right) - mu\right), KbT\right)\right)\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(Ev + Vef\right), mu\right), KbT\right)\right)\right)\right) \]

       6. +-lowering-+.f64 51.8%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(Ev, Vef\right), mu\right), KbT\right)\right)\right)\right) \]

   12. Simplified 51.8%

       \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.65 \cdot 10^{+122}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ \mathbf{elif}\;KbT \leq 4.6 \cdot 10^{+196}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;mu \leq -5.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= mu -5.2e+196)
   (/ NdChar (+ (exp (/ mu KbT)) 1.0))
   (/ NaChar (+ (exp (/ (+ (+ EAccept Ev) (- Vef mu)) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= -5.2e+196) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (mu <= (-5.2d+196)) then
        tmp = ndchar / (exp((mu / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp((((eaccept + ev) + (vef - mu)) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= -5.2e+196) {
		tmp = NdChar / (Math.exp((mu / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if mu <= -5.2e+196:
		tmp = NdChar / (math.exp((mu / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (mu <= -5.2e+196)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (mu <= -5.2e+196)
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	else
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[mu, -5.2e+196], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if mu < -5.20000000000000024e196

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 74.4%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in mu around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{mu}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 71.6%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(mu, KbT\right)\right)\right)\right) \]

   10. Simplified 71.6%

       \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{mu}{KbT}}}} \]

   if -5.20000000000000024e196 < 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%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 62.3%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -5.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;NdChar \leq -3.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{elif}\;NdChar \leq 1.9 \cdot 10^{+78}:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0)))
   (if (<= NdChar -3.6e-90)
     (/ NdChar t_0)
     (if (<= NdChar 1.9e+78)
       (/ NaChar t_0)
       (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((Vef / KbT)) + 1.0;
	double tmp;
	if (NdChar <= -3.6e-90) {
		tmp = NdChar / t_0;
	} else if (NdChar <= 1.9e+78) {
		tmp = NaChar / t_0;
	} else {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    if (ndchar <= (-3.6d-90)) then
        tmp = ndchar / t_0
    else if (ndchar <= 1.9d+78) then
        tmp = nachar / t_0
    else
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp((Vef / KbT)) + 1.0;
	double tmp;
	if (NdChar <= -3.6e-90) {
		tmp = NdChar / t_0;
	} else if (NdChar <= 1.9e+78) {
		tmp = NaChar / t_0;
	} else {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	tmp = 0
	if NdChar <= -3.6e-90:
		tmp = NdChar / t_0
	elif NdChar <= 1.9e+78:
		tmp = NaChar / t_0
	else:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	tmp = 0.0
	if (NdChar <= -3.6e-90)
		tmp = Float64(NdChar / t_0);
	elseif (NdChar <= 1.9e+78)
		tmp = Float64(NaChar / t_0);
	else
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp((Vef / KbT)) + 1.0;
	tmp = 0.0;
	if (NdChar <= -3.6e-90)
		tmp = NdChar / t_0;
	elseif (NdChar <= 1.9e+78)
		tmp = NaChar / t_0;
	else
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[NdChar, -3.6e-90], N[(NdChar / t$95$0), $MachinePrecision], If[LessEqual[NdChar, 1.9e+78], N[(NaChar / t$95$0), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -3.59999999999999981e-90

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 66.6%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 48.7%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 48.7%

       \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if -3.59999999999999981e-90 < NdChar < 1.9e78

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 71.4%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 49.3%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   12. Simplified 49.3%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if 1.9e78 < 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%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 77.9%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in EDonor around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EDonor}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 51.6%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EDonor, KbT\right)\right)\right)\right) \]

   10. Simplified 51.6%

       \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -3.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 1.9 \cdot 10^{+78}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -1.75 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq 9.2 \cdot 10^{-297}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -1.75e+75)
   (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
   (if (<= Ev 9.2e-297)
     (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
     (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -1.75e+75) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else if (Ev <= 9.2e-297) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ev <= (-1.75d+75)) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else if (ev <= 9.2d-297) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -1.75e+75) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else if (Ev <= 9.2e-297) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -1.75e+75:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	elif Ev <= 9.2e-297:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -1.75e+75)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	elseif (Ev <= 9.2e-297)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -1.75e+75)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	elseif (Ev <= 9.2e-297)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -1.75e+75], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, 9.2e-297], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Ev < -1.7499999999999999e75

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 58.0%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in Ev around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 49.7%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right)\right)\right)\right) \]

   12. Simplified 49.7%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   if -1.7499999999999999e75 < Ev < 9.1999999999999996e-297

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 61.5%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 43.5%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   12. Simplified 43.5%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if 9.1999999999999996e-297 < Ev

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 57.5%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in EAccept around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 38.0%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right)\right)\right)\right) \]

   12. Simplified 38.0%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -1.75 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;Ev \leq 9.2 \cdot 10^{-297}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -4.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -4.8e+72)
   (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
   (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -4.8e+72) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ev <= (-4.8d+72)) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -4.8e+72) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -4.8e+72:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -4.8e+72)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -4.8e+72)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	else
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -4.8e+72], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Ev < -4.8000000000000002e72

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 58.8%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 58.8%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in Ev around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 50.7%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right)\right)\right)\right) \]

   12. Simplified 50.7%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   if -4.8000000000000002e72 < 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%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 59.0%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 59.0%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in EAccept around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 38.4%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right)\right)\right)\right) \]

   12. Simplified 38.4%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -4.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ t_1 := \left(EAccept + \left(Vef + Ev\right)\right) - mu\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;KbT \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{t\_1 + \frac{0.16666666666666666 \cdot \frac{t\_1 \cdot t\_2}{KbT} + t\_2 \cdot 0.5}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar 2.0)
          (/
           NaChar
           (+
            2.0
            (/
             (+
              (+ (+ EAccept Ev) (- Vef mu))
              (*
               -0.5
               (*
                (+ EAccept (+ Ev (- Vef mu)))
                (/ (- (- (- mu Vef) Ev) EAccept) KbT))))
             KbT)))))
        (t_1 (- (+ EAccept (+ Vef Ev)) mu))
        (t_2 (* t_1 t_1)))
   (if (<= KbT -1.25e+109)
     t_0
     (if (<= KbT 2.4e+58)
       (/
        NaChar
        (+
         2.0
         (/
          (+
           t_1
           (/ (+ (* 0.16666666666666666 (/ (* t_1 t_2) KbT)) (* t_2 0.5)) KbT))
          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 / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	double t_1 = (EAccept + (Vef + Ev)) - mu;
	double t_2 = t_1 * t_1;
	double tmp;
	if (KbT <= -1.25e+109) {
		tmp = t_0;
	} else if (KbT <= 2.4e+58) {
		tmp = NaChar / (2.0 + ((t_1 + (((0.16666666666666666 * ((t_1 * t_2) / KbT)) + (t_2 * 0.5)) / KbT)) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (ndchar / 2.0d0) + (nachar / (2.0d0 + ((((eaccept + ev) + (vef - mu)) + ((-0.5d0) * ((eaccept + (ev + (vef - mu))) * ((((mu - vef) - ev) - eaccept) / kbt)))) / kbt)))
    t_1 = (eaccept + (vef + ev)) - mu
    t_2 = t_1 * t_1
    if (kbt <= (-1.25d+109)) then
        tmp = t_0
    else if (kbt <= 2.4d+58) then
        tmp = nachar / (2.0d0 + ((t_1 + (((0.16666666666666666d0 * ((t_1 * t_2) / kbt)) + (t_2 * 0.5d0)) / kbt)) / 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 / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	double t_1 = (EAccept + (Vef + Ev)) - mu;
	double t_2 = t_1 * t_1;
	double tmp;
	if (KbT <= -1.25e+109) {
		tmp = t_0;
	} else if (KbT <= 2.4e+58) {
		tmp = NaChar / (2.0 + ((t_1 + (((0.16666666666666666 * ((t_1 * t_2) / KbT)) + (t_2 * 0.5)) / KbT)) / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)))
	t_1 = (EAccept + (Vef + Ev)) - mu
	t_2 = t_1 * t_1
	tmp = 0
	if KbT <= -1.25e+109:
		tmp = t_0
	elif KbT <= 2.4e+58:
		tmp = NaChar / (2.0 + ((t_1 + (((0.16666666666666666 * ((t_1 * t_2) / KbT)) + (t_2 * 0.5)) / KbT)) / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) + Float64(-0.5 * Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) * Float64(Float64(Float64(Float64(mu - Vef) - Ev) - EAccept) / KbT)))) / KbT))))
	t_1 = Float64(Float64(EAccept + Float64(Vef + Ev)) - mu)
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (KbT <= -1.25e+109)
		tmp = t_0;
	elseif (KbT <= 2.4e+58)
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(t_1 + Float64(Float64(Float64(0.16666666666666666 * Float64(Float64(t_1 * t_2) / KbT)) + Float64(t_2 * 0.5)) / KbT)) / 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 / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	t_1 = (EAccept + (Vef + Ev)) - mu;
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (KbT <= -1.25e+109)
		tmp = t_0;
	elseif (KbT <= 2.4e+58)
		tmp = NaChar / (2.0 + ((t_1 + (((0.16666666666666666 * ((t_1 * t_2) / KbT)) + (t_2 * 0.5)) / KbT)) / 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 / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(mu - Vef), $MachinePrecision] - Ev), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[KbT, -1.25e+109], t$95$0, If[LessEqual[KbT, 2.4e+58], N[(NaChar / N[(2.0 + N[(N[(t$95$1 + N[(N[(N[(0.16666666666666666 * N[(N[(t$95$1 * t$95$2), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -1.25e109 or 2.4e58 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 + \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 - \color{blue}{\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}\right), \color{blue}{KbT}\right)\right)\right)\right) \]

   7. Simplified 69.3%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \color{blue}{2}\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right)\right), KbT\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 44.2%

       \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{2} \cdot \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT} \cdot \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       6. associate-+l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       11. associate-+l+ N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       14. --lowering--.f64 60.7%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

   12. Applied egg-rr 60.7%

       \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{2 - \frac{\color{blue}{\left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}\right) \cdot -0.5} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}} \]

   if -1.25e109 < KbT < 2.4e58

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 63.8%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in KbT around -inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right) \]

   12. Simplified 29.2%

       \[\leadsto \frac{NaChar}{\color{blue}{2 + \left(-\frac{-1 \cdot \left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{0.16666666666666666 \cdot \frac{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) \cdot \left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right)}{KbT} + 0.5 \cdot \left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right)}{KbT}\right)}{KbT}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ \mathbf{elif}\;KbT \leq 2.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) + \frac{0.16666666666666666 \cdot \frac{\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) \cdot \left(\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) \cdot \left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right)\right)}{KbT} + \left(\left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right) \cdot \left(\left(EAccept + \left(Vef + Ev\right)\right) - mu\right)\right) \cdot 0.5}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef + mu\right)\\ t_1 := \frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ t_2 := t\_0 - Ec\\ \mathbf{if}\;KbT \leq -3.5 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq 8.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{t\_2 + -0.5 \cdot \frac{t\_2 \cdot \left(Ec - t\_0\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (+ Vef mu)))
        (t_1
         (+
          (/ NdChar 2.0)
          (/
           NaChar
           (+
            2.0
            (/
             (+
              (+ (+ EAccept Ev) (- Vef mu))
              (*
               -0.5
               (*
                (+ EAccept (+ Ev (- Vef mu)))
                (/ (- (- (- mu Vef) Ev) EAccept) KbT))))
             KbT)))))
        (t_2 (- t_0 Ec)))
   (if (<= KbT -3.5e+145)
     t_1
     (if (<= KbT 8.8e-18)
       (/ NdChar (+ 2.0 (/ (+ t_2 (* -0.5 (/ (* t_2 (- Ec t_0)) KbT))) 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 = EDonor + (Vef + mu);
	double t_1 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	double t_2 = t_0 - Ec;
	double tmp;
	if (KbT <= -3.5e+145) {
		tmp = t_1;
	} else if (KbT <= 8.8e-18) {
		tmp = NdChar / (2.0 + ((t_2 + (-0.5 * ((t_2 * (Ec - t_0)) / KbT))) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = edonor + (vef + mu)
    t_1 = (ndchar / 2.0d0) + (nachar / (2.0d0 + ((((eaccept + ev) + (vef - mu)) + ((-0.5d0) * ((eaccept + (ev + (vef - mu))) * ((((mu - vef) - ev) - eaccept) / kbt)))) / kbt)))
    t_2 = t_0 - ec
    if (kbt <= (-3.5d+145)) then
        tmp = t_1
    else if (kbt <= 8.8d-18) then
        tmp = ndchar / (2.0d0 + ((t_2 + ((-0.5d0) * ((t_2 * (ec - t_0)) / kbt))) / kbt))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef + mu);
	double t_1 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	double t_2 = t_0 - Ec;
	double tmp;
	if (KbT <= -3.5e+145) {
		tmp = t_1;
	} else if (KbT <= 8.8e-18) {
		tmp = NdChar / (2.0 + ((t_2 + (-0.5 * ((t_2 * (Ec - t_0)) / KbT))) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = EDonor + (Vef + mu)
	t_1 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)))
	t_2 = t_0 - Ec
	tmp = 0
	if KbT <= -3.5e+145:
		tmp = t_1
	elif KbT <= 8.8e-18:
		tmp = NdChar / (2.0 + ((t_2 + (-0.5 * ((t_2 * (Ec - t_0)) / KbT))) / KbT))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef + mu))
	t_1 = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) + Float64(-0.5 * Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) * Float64(Float64(Float64(Float64(mu - Vef) - Ev) - EAccept) / KbT)))) / KbT))))
	t_2 = Float64(t_0 - Ec)
	tmp = 0.0
	if (KbT <= -3.5e+145)
		tmp = t_1;
	elseif (KbT <= 8.8e-18)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(t_2 + Float64(-0.5 * Float64(Float64(t_2 * Float64(Ec - t_0)) / KbT))) / KbT)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = EDonor + (Vef + mu);
	t_1 = (NdChar / 2.0) + (NaChar / (2.0 + ((((EAccept + Ev) + (Vef - mu)) + (-0.5 * ((EAccept + (Ev + (Vef - mu))) * ((((mu - Vef) - Ev) - EAccept) / KbT)))) / KbT)));
	t_2 = t_0 - Ec;
	tmp = 0.0;
	if (KbT <= -3.5e+145)
		tmp = t_1;
	elseif (KbT <= 8.8e-18)
		tmp = NdChar / (2.0 + ((t_2 + (-0.5 * ((t_2 * (Ec - t_0)) / KbT))) / KbT));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(mu - Vef), $MachinePrecision] - Ev), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - Ec), $MachinePrecision]}, If[LessEqual[KbT, -3.5e+145], t$95$1, If[LessEqual[KbT, 8.8e-18], N[(NdChar / N[(2.0 + N[(N[(t$95$2 + N[(-0.5 * N[(N[(t$95$2 * N[(Ec - t$95$0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -3.5000000000000001e145 or 8.7999999999999994e-18 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 + \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 - \color{blue}{\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}\right), \color{blue}{KbT}\right)\right)\right)\right) \]

   7. Simplified 70.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \color{blue}{2}\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right)\right), KbT\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 40.7%

       \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{2} \cdot \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT} \cdot \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       6. associate-+l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \left(\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       11. associate-+l+ N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

       14. --lowering--.f64 55.7%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]

   12. Applied egg-rr 55.7%

       \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{2 - \frac{\color{blue}{\left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}\right) \cdot -0.5} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}} \]

   if -3.5000000000000001e145 < KbT < 8.7999999999999994e-18

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 62.9%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right) \]

   10. Simplified 30.2%

       \[\leadsto \frac{NdChar}{\color{blue}{2 + \left(-\frac{\left(-\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right) + -0.5 \cdot \frac{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}}{KbT}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ \mathbf{elif}\;KbT \leq 8.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + -0.5 \cdot \frac{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) + -0.5 \cdot \left(\left(EAccept + \left(Ev + \left(Vef - mu\right)\right)\right) \cdot \frac{\left(\left(mu - Vef\right) - Ev\right) - EAccept}{KbT}\right)}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef + mu\right)\\ t_1 := t\_0 - Ec\\ t_2 := \frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \mathbf{if}\;KbT \leq -7.8 \cdot 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;KbT \leq 2.16 \cdot 10^{+101}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{t\_1 + -0.5 \cdot \frac{t\_1 \cdot \left(Ec - t\_0\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (+ Vef mu)))
        (t_1 (- t_0 Ec))
        (t_2
         (+
          (/ NdChar 2.0)
          (/ NaChar (+ 2.0 (/ (- (+ EAccept (+ Vef Ev)) mu) KbT))))))
   (if (<= KbT -7.8e+144)
     t_2
     (if (<= KbT 2.16e+101)
       (/ NdChar (+ 2.0 (/ (+ t_1 (* -0.5 (/ (* t_1 (- Ec t_0)) KbT))) KbT)))
       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 = EDonor + (Vef + mu);
	double t_1 = t_0 - Ec;
	double t_2 = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	double tmp;
	if (KbT <= -7.8e+144) {
		tmp = t_2;
	} else if (KbT <= 2.16e+101) {
		tmp = NdChar / (2.0 + ((t_1 + (-0.5 * ((t_1 * (Ec - t_0)) / KbT))) / KbT));
	} 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 = edonor + (vef + mu)
    t_1 = t_0 - ec
    t_2 = (ndchar / 2.0d0) + (nachar / (2.0d0 + (((eaccept + (vef + ev)) - mu) / kbt)))
    if (kbt <= (-7.8d+144)) then
        tmp = t_2
    else if (kbt <= 2.16d+101) then
        tmp = ndchar / (2.0d0 + ((t_1 + ((-0.5d0) * ((t_1 * (ec - t_0)) / kbt))) / kbt))
    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 = EDonor + (Vef + mu);
	double t_1 = t_0 - Ec;
	double t_2 = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	double tmp;
	if (KbT <= -7.8e+144) {
		tmp = t_2;
	} else if (KbT <= 2.16e+101) {
		tmp = NdChar / (2.0 + ((t_1 + (-0.5 * ((t_1 * (Ec - t_0)) / KbT))) / KbT));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = EDonor + (Vef + mu)
	t_1 = t_0 - Ec
	t_2 = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)))
	tmp = 0
	if KbT <= -7.8e+144:
		tmp = t_2
	elif KbT <= 2.16e+101:
		tmp = NdChar / (2.0 + ((t_1 + (-0.5 * ((t_1 * (Ec - t_0)) / KbT))) / KbT))
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef + mu))
	t_1 = Float64(t_0 - Ec)
	t_2 = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT))))
	tmp = 0.0
	if (KbT <= -7.8e+144)
		tmp = t_2;
	elseif (KbT <= 2.16e+101)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(t_1 + Float64(-0.5 * Float64(Float64(t_1 * Float64(Ec - t_0)) / KbT))) / KbT)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = EDonor + (Vef + mu);
	t_1 = t_0 - Ec;
	t_2 = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	tmp = 0.0;
	if (KbT <= -7.8e+144)
		tmp = t_2;
	elseif (KbT <= 2.16e+101)
		tmp = NdChar / (2.0 + ((t_1 + (-0.5 * ((t_1 * (Ec - t_0)) / KbT))) / KbT));
	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[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - Ec), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -7.8e+144], t$95$2, If[LessEqual[KbT, 2.16e+101], N[(NdChar / N[(2.0 + N[(N[(t$95$1 + N[(-0.5 * N[(N[(t$95$1 * N[(Ec - t$95$0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -7.80000000000000036e144 or 2.15999999999999999e101 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 + \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 - \color{blue}{\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}\right), \color{blue}{KbT}\right)\right)\right)\right) \]

   7. Simplified 67.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \color{blue}{2}\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right)\right), KbT\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 44.8%

       \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}} \]

   11. Taylor expanded in KbT around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{mu - \left(EAccept + \left(Ev + Vef\right)\right)}{KbT}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(mu - \left(EAccept + \left(Ev + Vef\right)\right)\right), \color{blue}{KbT}\right)\right)\right)\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \left(EAccept + \left(Ev + Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

       4. +-lowering-+.f64 62.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

   13. Simplified 62.0%

       \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{2 - \color{blue}{\frac{mu - \left(EAccept + \left(Ev + Vef\right)\right)}{KbT}}} \]

   if -7.80000000000000036e144 < KbT < 2.15999999999999999e101

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 64.6%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 64.6%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right) \]

   10. Simplified 29.3%

       \[\leadsto \frac{NdChar}{\color{blue}{2 + \left(-\frac{\left(-\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right) + -0.5 \cdot \frac{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}}{KbT}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -7.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 2.16 \cdot 10^{+101}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + -0.5 \cdot \frac{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \mathbf{if}\;KbT \leq -5.8 \cdot 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq -1.85 \cdot 10^{-193}:\\ \;\;\;\;\frac{NaChar}{2 + \left(\frac{Ev}{KbT} + \frac{0.5 \cdot \left(Ev \cdot Ev\right)}{KbT \cdot KbT}\right)}\\ \mathbf{elif}\;KbT \leq 3.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar 2.0)
          (/ NaChar (+ 2.0 (/ (- (+ EAccept (+ Vef Ev)) mu) KbT))))))
   (if (<= KbT -5.8e+115)
     t_0
     (if (<= KbT -1.85e-193)
       (/ NaChar (+ 2.0 (+ (/ Ev KbT) (/ (* 0.5 (* Ev Ev)) (* KbT KbT)))))
       (if (<= KbT 3.1e-18)
         (/
          NaChar
          (-
           (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Vef KbT) (/ Ev KbT)))
           (/ 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 / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	double tmp;
	if (KbT <= -5.8e+115) {
		tmp = t_0;
	} else if (KbT <= -1.85e-193) {
		tmp = NaChar / (2.0 + ((Ev / KbT) + ((0.5 * (Ev * Ev)) / (KbT * KbT))));
	} else if (KbT <= 3.1e-18) {
		tmp = NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / 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 / 2.0d0) + (nachar / (2.0d0 + (((eaccept + (vef + ev)) - mu) / kbt)))
    if (kbt <= (-5.8d+115)) then
        tmp = t_0
    else if (kbt <= (-1.85d-193)) then
        tmp = nachar / (2.0d0 + ((ev / kbt) + ((0.5d0 * (ev * ev)) / (kbt * kbt))))
    else if (kbt <= 3.1d-18) then
        tmp = nachar / (((2.0d0 + (eaccept / kbt)) + ((vef / kbt) + (ev / kbt))) - (mu / 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 / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	double tmp;
	if (KbT <= -5.8e+115) {
		tmp = t_0;
	} else if (KbT <= -1.85e-193) {
		tmp = NaChar / (2.0 + ((Ev / KbT) + ((0.5 * (Ev * Ev)) / (KbT * KbT))));
	} else if (KbT <= 3.1e-18) {
		tmp = NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)))
	tmp = 0
	if KbT <= -5.8e+115:
		tmp = t_0
	elif KbT <= -1.85e-193:
		tmp = NaChar / (2.0 + ((Ev / KbT) + ((0.5 * (Ev * Ev)) / (KbT * KbT))))
	elif KbT <= 3.1e-18:
		tmp = NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT))))
	tmp = 0.0
	if (KbT <= -5.8e+115)
		tmp = t_0;
	elseif (KbT <= -1.85e-193)
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(Ev / KbT) + Float64(Float64(0.5 * Float64(Ev * Ev)) / Float64(KbT * KbT)))));
	elseif (KbT <= 3.1e-18)
		tmp = Float64(NaChar / Float64(Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) - Float64(mu / 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 / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	tmp = 0.0;
	if (KbT <= -5.8e+115)
		tmp = t_0;
	elseif (KbT <= -1.85e-193)
		tmp = NaChar / (2.0 + ((Ev / KbT) + ((0.5 * (Ev * Ev)) / (KbT * KbT))));
	elseif (KbT <= 3.1e-18)
		tmp = NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / 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 / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -5.8e+115], t$95$0, If[LessEqual[KbT, -1.85e-193], N[(NaChar / N[(2.0 + N[(N[(Ev / KbT), $MachinePrecision] + N[(N[(0.5 * N[(Ev * Ev), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 3.1e-18], N[(NaChar / N[(N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -5.80000000000000009e115 or 3.10000000000000007e-18 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 + \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 - \color{blue}{\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}\right), \color{blue}{KbT}\right)\right)\right)\right) \]

   7. Simplified 72.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \color{blue}{2}\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right)\right), KbT\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 42.0%

       \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}} \]

   11. Taylor expanded in KbT around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{mu - \left(EAccept + \left(Ev + Vef\right)\right)}{KbT}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(mu - \left(EAccept + \left(Ev + Vef\right)\right)\right), \color{blue}{KbT}\right)\right)\right)\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \left(EAccept + \left(Ev + Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

       4. +-lowering-+.f64 52.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

   13. Simplified 52.4%

       \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{2 - \color{blue}{\frac{mu - \left(EAccept + \left(Ev + Vef\right)\right)}{KbT}}} \]

   if -5.80000000000000009e115 < KbT < -1.8500000000000001e-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%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 56.4%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in Ev around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 28.9%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right)\right)\right)\right) \]

   12. Simplified 28.9%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   13. Taylor expanded in KbT around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + \left(\frac{1}{2} \cdot \frac{{Ev}^{2}}{{KbT}^{2}} + \frac{Ev}{KbT}\right)\right)}\right) \]
   14. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot \frac{{Ev}^{2}}{{KbT}^{2}} + \frac{Ev}{KbT}\right)}\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \left(\frac{Ev}{KbT} + \color{blue}{\frac{1}{2} \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{Ev}{KbT}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \left(\frac{\frac{1}{2} \cdot {Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {Ev}^{2}\right), \color{blue}{\left({KbT}^{2}\right)}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({Ev}^{2}\right)\right), \left({\color{blue}{KbT}}^{2}\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(Ev \cdot Ev\right)\right), \left({KbT}^{2}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(Ev, Ev\right)\right), \left({KbT}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(Ev, Ev\right)\right), \left(KbT \cdot \color{blue}{KbT}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 14.7%

           \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(Ev, Ev\right)\right), \mathsf{*.f64}\left(KbT, \color{blue}{KbT}\right)\right)\right)\right)\right) \]

   15. Simplified 14.7%

       \[\leadsto \frac{NaChar}{\color{blue}{2 + \left(\frac{Ev}{KbT} + \frac{0.5 \cdot \left(Ev \cdot Ev\right)}{KbT \cdot KbT}\right)}} \]

   if -1.8500000000000001e-193 < KbT < 3.10000000000000007e-18

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 73.9%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 73.9%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in KbT around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\right) \]
   11. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right)\right), \color{blue}{\left(\frac{mu}{KbT}\right)}\right)\right) \]

       2. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right), \left(\frac{\color{blue}{mu}}{KbT}\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(2 + \frac{EAccept}{KbT}\right), \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right), \left(\frac{\color{blue}{mu}}{KbT}\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EAccept}{KbT}\right)\right), \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Ev}{KbT}\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \left(\frac{Vef}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right) \]

       9. /-lowering-/.f64 26.4%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right), \mathsf{/.f64}\left(Vef, KbT\right)\right)\right), \mathsf{/.f64}\left(mu, \color{blue}{KbT}\right)\right)\right) \]

   12. Simplified 26.4%

       \[\leadsto \frac{NaChar}{\color{blue}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \mathbf{elif}\;KbT \leq -1.85 \cdot 10^{-193}:\\ \;\;\;\;\frac{NaChar}{2 + \left(\frac{Ev}{KbT} + \frac{0.5 \cdot \left(Ev \cdot Ev\right)}{KbT \cdot KbT}\right)}\\ \mathbf{elif}\;KbT \leq 3.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 33.1% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4.1 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.28 \cdot 10^{-18}:\\ \;\;\;\;\frac{NdChar}{\frac{mu \cdot \left(\frac{EDonor + \left(Vef + \left(KbT \cdot 2 - Ec\right)\right)}{mu} + 1\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4.1e-10)
   (* 0.5 (+ NdChar NaChar))
   (if (<= KbT 1.28e-18)
     (/
      NdChar
      (/ (* mu (+ (/ (+ EDonor (+ Vef (- (* KbT 2.0) Ec))) mu) 1.0)) KbT))
     (+
      (/ NdChar 2.0)
      (/ NaChar (+ 2.0 (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.1e-10) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 1.28e-18) {
		tmp = NdChar / ((mu * (((EDonor + (Vef + ((KbT * 2.0) - Ec))) / mu) + 1.0)) / KbT);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-4.1d-10)) then
        tmp = 0.5d0 * (ndchar + nachar)
    else if (kbt <= 1.28d-18) then
        tmp = ndchar / ((mu * (((edonor + (vef + ((kbt * 2.0d0) - ec))) / mu) + 1.0d0)) / kbt)
    else
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 + (((eaccept + (vef + ev)) - mu) / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.1e-10) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 1.28e-18) {
		tmp = NdChar / ((mu * (((EDonor + (Vef + ((KbT * 2.0) - Ec))) / mu) + 1.0)) / KbT);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -4.1e-10:
		tmp = 0.5 * (NdChar + NaChar)
	elif KbT <= 1.28e-18:
		tmp = NdChar / ((mu * (((EDonor + (Vef + ((KbT * 2.0) - Ec))) / mu) + 1.0)) / KbT)
	else:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4.1e-10)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	elseif (KbT <= 1.28e-18)
		tmp = Float64(NdChar / Float64(Float64(mu * Float64(Float64(Float64(EDonor + Float64(Vef + Float64(Float64(KbT * 2.0) - Ec))) / mu) + 1.0)) / KbT));
	else
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -4.1e-10)
		tmp = 0.5 * (NdChar + NaChar);
	elseif (KbT <= 1.28e-18)
		tmp = NdChar / ((mu * (((EDonor + (Vef + ((KbT * 2.0) - Ec))) / mu) + 1.0)) / KbT);
	else
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4.1e-10], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.28e-18], N[(NdChar / N[(N[(mu * N[(N[(N[(EDonor + N[(Vef + N[(N[(KbT * 2.0), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.0999999999999998e-10

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 44.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 44.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -4.0999999999999998e-10 < KbT < 1.27999999999999993e-18

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 62.2%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right), \color{blue}{\left(\frac{Ec}{KbT}\right)}\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(2 + \frac{EDonor}{KbT}\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EDonor}{KbT}\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      9. /-lowering-/.f64 17.3%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \mathsf{/.f64}\left(Ec, \color{blue}{KbT}\right)\right)\right) \]

   10. Simplified 17.3%

       \[\leadsto \frac{NdChar}{\color{blue}{\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}} \]

   11. Taylor expanded in KbT around 0

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\frac{\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec\right), \color{blue}{KbT}\right)\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec\right)\right), KbT\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec\right)\right), KbT\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\left(\left(Vef + mu\right) + 2 \cdot KbT\right), Ec\right)\right), KbT\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + mu\right), \left(2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, mu\right), \left(2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       8. *-lowering-*.f64 22.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, mu\right), \mathsf{*.f64}\left(2, KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

   13. Simplified 22.6%

       \[\leadsto \frac{NdChar}{\color{blue}{\frac{EDonor + \left(\left(\left(Vef + mu\right) + 2 \cdot KbT\right) - Ec\right)}{KbT}}} \]

   14. Taylor expanded in mu around -inf

       \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(mu \cdot \left(-1 \cdot \frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu} - 1\right)\right)\right)}, KbT\right)\right) \]
   15. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(\left(-1 \cdot mu\right) \cdot \left(-1 \cdot \frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu} - 1\right)\right), KbT\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot mu\right), \left(-1 \cdot \frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu} - 1\right)\right), KbT\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(mu\right)\right), \left(-1 \cdot \frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu} - 1\right)\right), KbT\right)\right) \]

       4. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \left(-1 \cdot \frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu} - 1\right)\right), KbT\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \left(-1 \cdot \frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), KbT\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \left(-1 \cdot \frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu} + -1\right)\right), KbT\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\left(-1 \cdot \frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu}\right), -1\right)\right), KbT\right)\right) \]

       8. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu}\right)\right), -1\right)\right), KbT\right)\right) \]

       9. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\left(\frac{\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec}{mu}\right)\right), -1\right)\right), KbT\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + 2 \cdot KbT\right)\right) - Ec\right), mu\right)\right), -1\right)\right), KbT\right)\right) \]

       11. associate--l+ N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + 2 \cdot KbT\right) - Ec\right)\right), mu\right)\right), -1\right)\right), KbT\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + 2 \cdot KbT\right) - Ec\right)\right), mu\right)\right), -1\right)\right), KbT\right)\right) \]

       13. associate--l+ N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(2 \cdot KbT - Ec\right)\right)\right), mu\right)\right), -1\right)\right), KbT\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(2 \cdot KbT - Ec\right)\right)\right), mu\right)\right), -1\right)\right), KbT\right)\right) \]

       15. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(2 \cdot KbT\right), Ec\right)\right)\right), mu\right)\right), -1\right)\right), KbT\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(KbT \cdot 2\right), Ec\right)\right)\right), mu\right)\right), -1\right)\right), KbT\right)\right) \]

       17. *-lowering-*.f64 25.0%

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(mu\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(KbT, 2\right), Ec\right)\right)\right), mu\right)\right), -1\right)\right), KbT\right)\right) \]

   16. Simplified 25.0%

       \[\leadsto \frac{NdChar}{\frac{\color{blue}{\left(-mu\right) \cdot \left(\left(-\frac{EDonor + \left(Vef + \left(KbT \cdot 2 - Ec\right)\right)}{mu}\right) + -1\right)}}{KbT}} \]

   if 1.27999999999999993e-18 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 + \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 - \color{blue}{\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}\right), \color{blue}{KbT}\right)\right)\right)\right) \]

   7. Simplified 71.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \color{blue}{2}\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right)\right), KbT\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 38.0%

       \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}} \]

   11. Taylor expanded in KbT around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{mu - \left(EAccept + \left(Ev + Vef\right)\right)}{KbT}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(mu - \left(EAccept + \left(Ev + Vef\right)\right)\right), \color{blue}{KbT}\right)\right)\right)\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \left(EAccept + \left(Ev + Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

       4. +-lowering-+.f64 45.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

   13. Simplified 45.1%

       \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{2 - \color{blue}{\frac{mu - \left(EAccept + \left(Ev + Vef\right)\right)}{KbT}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.1 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.28 \cdot 10^{-18}:\\ \;\;\;\;\frac{NdChar}{\frac{mu \cdot \left(\frac{EDonor + \left(Vef + \left(KbT \cdot 2 - Ec\right)\right)}{mu} + 1\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 32.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -8.5 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{NdChar}{\frac{EDonor + \left(\left(\left(Vef + mu\right) + KbT \cdot 2\right) - Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -8.5e-11)
   (* 0.5 (+ NdChar NaChar))
   (if (<= KbT 2.3e-19)
     (/ NdChar (/ (+ EDonor (- (+ (+ Vef mu) (* KbT 2.0)) Ec)) KbT))
     (+
      (/ NdChar 2.0)
      (/ NaChar (+ 2.0 (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -8.5e-11) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 2.3e-19) {
		tmp = NdChar / ((EDonor + (((Vef + mu) + (KbT * 2.0)) - Ec)) / KbT);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-8.5d-11)) then
        tmp = 0.5d0 * (ndchar + nachar)
    else if (kbt <= 2.3d-19) then
        tmp = ndchar / ((edonor + (((vef + mu) + (kbt * 2.0d0)) - ec)) / kbt)
    else
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 + (((eaccept + (vef + ev)) - mu) / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -8.5e-11) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 2.3e-19) {
		tmp = NdChar / ((EDonor + (((Vef + mu) + (KbT * 2.0)) - Ec)) / KbT);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -8.5e-11:
		tmp = 0.5 * (NdChar + NaChar)
	elif KbT <= 2.3e-19:
		tmp = NdChar / ((EDonor + (((Vef + mu) + (KbT * 2.0)) - Ec)) / KbT)
	else:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -8.5e-11)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	elseif (KbT <= 2.3e-19)
		tmp = Float64(NdChar / Float64(Float64(EDonor + Float64(Float64(Float64(Vef + mu) + Float64(KbT * 2.0)) - Ec)) / KbT));
	else
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -8.5e-11)
		tmp = 0.5 * (NdChar + NaChar);
	elseif (KbT <= 2.3e-19)
		tmp = NdChar / ((EDonor + (((Vef + mu) + (KbT * 2.0)) - Ec)) / KbT);
	else
		tmp = (NdChar / 2.0) + (NaChar / (2.0 + (((EAccept + (Vef + Ev)) - mu) / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -8.5e-11], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.3e-19], N[(NdChar / N[(N[(EDonor + N[(N[(N[(Vef + mu), $MachinePrecision] + N[(KbT * 2.0), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -8.50000000000000037e-11

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 44.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 44.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -8.50000000000000037e-11 < KbT < 2.2999999999999998e-19

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 62.2%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right), \color{blue}{\left(\frac{Ec}{KbT}\right)}\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(2 + \frac{EDonor}{KbT}\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EDonor}{KbT}\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      9. /-lowering-/.f64 17.3%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \mathsf{/.f64}\left(Ec, \color{blue}{KbT}\right)\right)\right) \]

   10. Simplified 17.3%

       \[\leadsto \frac{NdChar}{\color{blue}{\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}} \]

   11. Taylor expanded in KbT around 0

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\frac{\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec\right), \color{blue}{KbT}\right)\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec\right)\right), KbT\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec\right)\right), KbT\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\left(\left(Vef + mu\right) + 2 \cdot KbT\right), Ec\right)\right), KbT\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + mu\right), \left(2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, mu\right), \left(2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       8. *-lowering-*.f64 22.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, mu\right), \mathsf{*.f64}\left(2, KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

   13. Simplified 22.6%

       \[\leadsto \frac{NdChar}{\color{blue}{\frac{EDonor + \left(\left(\left(Vef + mu\right) + 2 \cdot KbT\right) - Ec\right)}{KbT}}} \]

   if 2.2999999999999998e-19 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 + \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \left(2 - \color{blue}{\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}^{2}}{KbT}\right), \color{blue}{KbT}\right)\right)\right)\right) \]

   7. Simplified 71.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \color{blue}{2}\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right)\right), KbT\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 38.0%

       \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right) \cdot \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)\right)}{KbT} - \left(\left(EAccept + Ev\right) + \left(Vef - mu\right)\right)}{KbT}} \]

   11. Taylor expanded in KbT around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{mu - \left(EAccept + \left(Ev + Vef\right)\right)}{KbT}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(mu - \left(EAccept + \left(Ev + Vef\right)\right)\right), \color{blue}{KbT}\right)\right)\right)\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \left(EAccept + \left(Ev + Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

       4. +-lowering-+.f64 45.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(mu, \mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right)\right), KbT\right)\right)\right)\right) \]

   13. Simplified 45.1%

       \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{2 - \color{blue}{\frac{mu - \left(EAccept + \left(Ev + Vef\right)\right)}{KbT}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -8.5 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{NdChar}{\frac{EDonor + \left(\left(\left(Vef + mu\right) + KbT \cdot 2\right) - Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 + \frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 33.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.05 \cdot 10^{-27}:\\ \;\;\;\;\frac{NdChar}{\frac{EDonor + \left(\left(\left(Vef + mu\right) + KbT \cdot 2\right) - Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -5e-11)
     t_0
     (if (<= KbT 2.05e-27)
       (/ NdChar (/ (+ EDonor (- (+ (+ Vef mu) (* KbT 2.0)) 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -5e-11) {
		tmp = t_0;
	} else if (KbT <= 2.05e-27) {
		tmp = NdChar / ((EDonor + (((Vef + mu) + (KbT * 2.0)) - 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 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-5d-11)) then
        tmp = t_0
    else if (kbt <= 2.05d-27) then
        tmp = ndchar / ((edonor + (((vef + mu) + (kbt * 2.0d0)) - 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -5e-11) {
		tmp = t_0;
	} else if (KbT <= 2.05e-27) {
		tmp = NdChar / ((EDonor + (((Vef + mu) + (KbT * 2.0)) - Ec)) / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -5e-11:
		tmp = t_0
	elif KbT <= 2.05e-27:
		tmp = NdChar / ((EDonor + (((Vef + mu) + (KbT * 2.0)) - Ec)) / KbT)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -5e-11)
		tmp = t_0;
	elseif (KbT <= 2.05e-27)
		tmp = Float64(NdChar / Float64(Float64(EDonor + Float64(Float64(Float64(Vef + mu) + Float64(KbT * 2.0)) - Ec)) / 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 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -5e-11)
		tmp = t_0;
	elseif (KbT <= 2.05e-27)
		tmp = NdChar / ((EDonor + (((Vef + mu) + (KbT * 2.0)) - 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[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -5e-11], t$95$0, If[LessEqual[KbT, 2.05e-27], N[(NdChar / N[(N[(EDonor + N[(N[(N[(Vef + mu), $MachinePrecision] + N[(KbT * 2.0), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -5.00000000000000018e-11 or 2.0499999999999999e-27 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 44.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 44.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -5.00000000000000018e-11 < KbT < 2.0499999999999999e-27

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 63.1%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right), \color{blue}{\left(\frac{Ec}{KbT}\right)}\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(2 + \frac{EDonor}{KbT}\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EDonor}{KbT}\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      9. /-lowering-/.f64 17.5%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \mathsf{/.f64}\left(Ec, \color{blue}{KbT}\right)\right)\right) \]

   10. Simplified 17.5%

       \[\leadsto \frac{NdChar}{\color{blue}{\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}} \]

   11. Taylor expanded in KbT around 0

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\frac{\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec\right), \color{blue}{KbT}\right)\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec\right)\right), KbT\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec\right)\right), KbT\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\left(\left(Vef + mu\right) + 2 \cdot KbT\right), Ec\right)\right), KbT\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + mu\right), \left(2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, mu\right), \left(2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       8. *-lowering-*.f64 23.0%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, mu\right), \mathsf{*.f64}\left(2, KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

   13. Simplified 23.0%

       \[\leadsto \frac{NdChar}{\color{blue}{\frac{EDonor + \left(\left(\left(Vef + mu\right) + 2 \cdot KbT\right) - Ec\right)}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.05 \cdot 10^{-27}:\\ \;\;\;\;\frac{NdChar}{\frac{EDonor + \left(\left(\left(Vef + mu\right) + KbT \cdot 2\right) - Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 32.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.12 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef + \left(\left(mu + KbT \cdot 2\right) - Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.12e-10)
     t_0
     (if (<= KbT 2.95e-25)
       (/ NdChar (/ (+ Vef (- (+ mu (* KbT 2.0)) 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.12e-10) {
		tmp = t_0;
	} else if (KbT <= 2.95e-25) {
		tmp = NdChar / ((Vef + ((mu + (KbT * 2.0)) - 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 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.12d-10)) then
        tmp = t_0
    else if (kbt <= 2.95d-25) then
        tmp = ndchar / ((vef + ((mu + (kbt * 2.0d0)) - 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.12e-10) {
		tmp = t_0;
	} else if (KbT <= 2.95e-25) {
		tmp = NdChar / ((Vef + ((mu + (KbT * 2.0)) - Ec)) / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.12e-10:
		tmp = t_0
	elif KbT <= 2.95e-25:
		tmp = NdChar / ((Vef + ((mu + (KbT * 2.0)) - Ec)) / KbT)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.12e-10)
		tmp = t_0;
	elseif (KbT <= 2.95e-25)
		tmp = Float64(NdChar / Float64(Float64(Vef + Float64(Float64(mu + Float64(KbT * 2.0)) - Ec)) / 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 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.12e-10)
		tmp = t_0;
	elseif (KbT <= 2.95e-25)
		tmp = NdChar / ((Vef + ((mu + (KbT * 2.0)) - 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[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.12e-10], t$95$0, If[LessEqual[KbT, 2.95e-25], N[(NdChar / N[(N[(Vef + N[(N[(mu + N[(KbT * 2.0), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -1.12e-10 or 2.9499999999999999e-25 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 44.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 44.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -1.12e-10 < KbT < 2.9499999999999999e-25

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 63.1%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right), \color{blue}{\left(\frac{Ec}{KbT}\right)}\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(2 + \frac{EDonor}{KbT}\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EDonor}{KbT}\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      9. /-lowering-/.f64 17.5%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \mathsf{/.f64}\left(Ec, \color{blue}{KbT}\right)\right)\right) \]

   10. Simplified 17.5%

       \[\leadsto \frac{NdChar}{\color{blue}{\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}} \]

   11. Taylor expanded in KbT around 0

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\frac{\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec\right), \color{blue}{KbT}\right)\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec\right)\right), KbT\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec\right)\right), KbT\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\left(\left(Vef + mu\right) + 2 \cdot KbT\right), Ec\right)\right), KbT\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + mu\right), \left(2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, mu\right), \left(2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       8. *-lowering-*.f64 23.0%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, mu\right), \mathsf{*.f64}\left(2, KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

   13. Simplified 23.0%

       \[\leadsto \frac{NdChar}{\color{blue}{\frac{EDonor + \left(\left(\left(Vef + mu\right) + 2 \cdot KbT\right) - Ec\right)}{KbT}}} \]

   14. Taylor expanded in EDonor around 0

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\frac{\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec}{KbT}\right)}\right) \]
   15. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(\left(Vef + \left(mu + 2 \cdot KbT\right)\right) - Ec\right), \color{blue}{KbT}\right)\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(Vef + \left(\left(mu + 2 \cdot KbT\right) - Ec\right)\right), KbT\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(\left(mu + 2 \cdot KbT\right) - Ec\right)\right), KbT\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + 2 \cdot KbT\right), Ec\right)\right), KbT\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, \left(2 \cdot KbT\right)\right), Ec\right)\right), KbT\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, \left(KbT \cdot 2\right)\right), Ec\right)\right), KbT\right)\right) \]

       7. *-lowering-*.f64 20.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, \mathsf{*.f64}\left(KbT, 2\right)\right), Ec\right)\right), KbT\right)\right) \]

   16. Simplified 20.6%

       \[\leadsto \frac{NdChar}{\color{blue}{\frac{Vef + \left(\left(mu + KbT \cdot 2\right) - Ec\right)}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.12 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef + \left(\left(mu + KbT \cdot 2\right) - Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 30.3% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -7 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.35 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{mu \cdot \left(NdChar \cdot KbT\right)}{EDonor}}{0 - EDonor}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -7e+30)
     t_0
     (if (<= KbT 2.35e-29)
       (/ (/ (* mu (* NdChar KbT)) EDonor) (- 0.0 EDonor))
       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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -7e+30) {
		tmp = t_0;
	} else if (KbT <= 2.35e-29) {
		tmp = ((mu * (NdChar * KbT)) / EDonor) / (0.0 - EDonor);
	} 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 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-7d+30)) then
        tmp = t_0
    else if (kbt <= 2.35d-29) then
        tmp = ((mu * (ndchar * kbt)) / edonor) / (0.0d0 - edonor)
    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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -7e+30) {
		tmp = t_0;
	} else if (KbT <= 2.35e-29) {
		tmp = ((mu * (NdChar * KbT)) / EDonor) / (0.0 - EDonor);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -7e+30:
		tmp = t_0
	elif KbT <= 2.35e-29:
		tmp = ((mu * (NdChar * KbT)) / EDonor) / (0.0 - EDonor)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -7e+30)
		tmp = t_0;
	elseif (KbT <= 2.35e-29)
		tmp = Float64(Float64(Float64(mu * Float64(NdChar * KbT)) / EDonor) / Float64(0.0 - EDonor));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -7e+30)
		tmp = t_0;
	elseif (KbT <= 2.35e-29)
		tmp = ((mu * (NdChar * KbT)) / EDonor) / (0.0 - EDonor);
	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[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -7e+30], t$95$0, If[LessEqual[KbT, 2.35e-29], N[(N[(N[(mu * N[(NdChar * KbT), $MachinePrecision]), $MachinePrecision] / EDonor), $MachinePrecision] / N[(0.0 - EDonor), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -7.00000000000000042e30 or 2.3499999999999999e-29 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 46.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 46.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -7.00000000000000042e30 < KbT < 2.3499999999999999e-29

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 62.7%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 62.7%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right), \color{blue}{\left(\frac{Ec}{KbT}\right)}\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(2 + \frac{EDonor}{KbT}\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EDonor}{KbT}\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      9. /-lowering-/.f64 17.2%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \mathsf{/.f64}\left(Ec, \color{blue}{KbT}\right)\right)\right) \]

   10. Simplified 17.2%

       \[\leadsto \frac{NdChar}{\color{blue}{\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}} \]

   11. Taylor expanded in EDonor around inf

       \[\leadsto \color{blue}{\frac{-1 \cdot \frac{{KbT}^{2} \cdot \left(NdChar \cdot \left(\left(2 + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}\right)\right)}{EDonor} + KbT \cdot NdChar}{EDonor}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{{KbT}^{2} \cdot \left(NdChar \cdot \left(\left(2 + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}\right)\right)}{EDonor} + KbT \cdot NdChar\right), \color{blue}{EDonor}\right) \]

   13. Simplified 4.3%

       \[\leadsto \color{blue}{\frac{\left(-\frac{\left(KbT \cdot KbT\right) \cdot \left(NdChar \cdot \left(2 + \left(\left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right) - \frac{Ec}{KbT}\right)\right)\right)}{EDonor}\right) + KbT \cdot NdChar}{EDonor}} \]

   14. Taylor expanded in mu around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{KbT \cdot \left(NdChar \cdot mu\right)}{EDonor}\right)}, EDonor\right) \]
   15. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{KbT \cdot \left(NdChar \cdot mu\right)}{EDonor}\right)\right), EDonor\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(KbT \cdot \left(NdChar \cdot mu\right)\right), EDonor\right)\right), EDonor\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(KbT \cdot NdChar\right) \cdot mu\right), EDonor\right)\right), EDonor\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(KbT \cdot NdChar\right), mu\right), EDonor\right)\right), EDonor\right) \]

       5. *-lowering-*.f64 20.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(KbT, NdChar\right), mu\right), EDonor\right)\right), EDonor\right) \]

   16. Simplified 20.1%

       \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\left(KbT \cdot NdChar\right) \cdot mu}{EDonor}}}{EDonor} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -7 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.35 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{mu \cdot \left(NdChar \cdot KbT\right)}{EDonor}}{0 - EDonor}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 26.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -6.4 \cdot 10^{+209}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;Ev \leq 9.8 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{Ec}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -6.4e+209)
   (/ NaChar (+ 2.0 (/ Ev KbT)))
   (if (<= Ev 9.8e-52)
     (* 0.5 (+ NdChar NaChar))
     (/ NdChar (- 2.0 (/ 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 (Ev <= -6.4e+209) {
		tmp = NaChar / (2.0 + (Ev / KbT));
	} else if (Ev <= 9.8e-52) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NdChar / (2.0 - (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 (ev <= (-6.4d+209)) then
        tmp = nachar / (2.0d0 + (ev / kbt))
    else if (ev <= 9.8d-52) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = ndchar / (2.0d0 - (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 (Ev <= -6.4e+209) {
		tmp = NaChar / (2.0 + (Ev / KbT));
	} else if (Ev <= 9.8e-52) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NdChar / (2.0 - (Ec / KbT));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -6.4e+209:
		tmp = NaChar / (2.0 + (Ev / KbT))
	elif Ev <= 9.8e-52:
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = NdChar / (2.0 - (Ec / KbT))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -6.4e+209)
		tmp = Float64(NaChar / Float64(2.0 + Float64(Ev / KbT)));
	elseif (Ev <= 9.8e-52)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(NdChar / Float64(2.0 - Float64(Ec / KbT)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -6.4e+209)
		tmp = NaChar / (2.0 + (Ev / KbT));
	elseif (Ev <= 9.8e-52)
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = NdChar / (2.0 - (Ec / KbT));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -6.4e+209], N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, 9.8e-52], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(2.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Ev < -6.3999999999999999e209

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 63.7%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in Ev around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 58.5%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right)\right)\right)\right) \]

   12. Simplified 58.5%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   13. Taylor expanded in Ev around 0

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + \frac{Ev}{KbT}\right)}\right) \]
   14. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right) \]

       2. /-lowering-/.f64 22.4%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(Ev, \color{blue}{KbT}\right)\right)\right) \]

   15. Simplified 22.4%

       \[\leadsto \frac{NaChar}{\color{blue}{2 + \frac{Ev}{KbT}}} \]

   if -6.3999999999999999e209 < Ev < 9.80000000000000037e-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%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 29.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 29.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if 9.80000000000000037e-52 < Ev

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 60.8%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right), \color{blue}{\left(\frac{Ec}{KbT}\right)}\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(2 + \frac{EDonor}{KbT}\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EDonor}{KbT}\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      9. /-lowering-/.f64 21.2%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \mathsf{/.f64}\left(Ec, \color{blue}{KbT}\right)\right)\right) \]

   10. Simplified 21.2%

       \[\leadsto \frac{NdChar}{\color{blue}{\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}} \]

   11. Taylor expanded in KbT around inf

       \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\color{blue}{2}, \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 20.1%

       \[\leadsto \frac{NdChar}{\color{blue}{2} - \frac{Ec}{KbT}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -6.4 \cdot 10^{+209}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;Ev \leq 9.8 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{Ec}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 29.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -3.6 \cdot 10^{-187}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.32 \cdot 10^{-28}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -3.6e-187)
     t_0
     (if (<= KbT 1.32e-28) (/ NdChar (/ Vef 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -3.6e-187) {
		tmp = t_0;
	} else if (KbT <= 1.32e-28) {
		tmp = NdChar / (Vef / 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 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-3.6d-187)) then
        tmp = t_0
    else if (kbt <= 1.32d-28) then
        tmp = ndchar / (vef / 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -3.6e-187) {
		tmp = t_0;
	} else if (KbT <= 1.32e-28) {
		tmp = NdChar / (Vef / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -3.6e-187:
		tmp = t_0
	elif KbT <= 1.32e-28:
		tmp = NdChar / (Vef / KbT)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -3.6e-187)
		tmp = t_0;
	elseif (KbT <= 1.32e-28)
		tmp = Float64(NdChar / Float64(Vef / 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 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -3.6e-187)
		tmp = t_0;
	elseif (KbT <= 1.32e-28)
		tmp = NdChar / (Vef / 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[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -3.6e-187], t$95$0, If[LessEqual[KbT, 1.32e-28], N[(NdChar / N[(Vef / KbT), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -3.59999999999999994e-187 or 1.32000000000000011e-28 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 36.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 36.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -3.59999999999999994e-187 < KbT < 1.32000000000000011e-28

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 58.4%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right), \color{blue}{\left(\frac{Ec}{KbT}\right)}\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(2 + \frac{EDonor}{KbT}\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{\color{blue}{Ec}}{KbT}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EDonor}{KbT}\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \left(\frac{mu}{KbT}\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \left(\frac{Ec}{KbT}\right)\right)\right) \]

      9. /-lowering-/.f64 18.2%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(mu, KbT\right)\right)\right), \mathsf{/.f64}\left(Ec, \color{blue}{KbT}\right)\right)\right) \]

   10. Simplified 18.2%

       \[\leadsto \frac{NdChar}{\color{blue}{\left(\left(2 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}} \]

   11. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\frac{Vef}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 19.3%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(Vef, \color{blue}{KbT}\right)\right) \]

   13. Simplified 19.3%

       \[\leadsto \frac{NdChar}{\color{blue}{\frac{Vef}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.6 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.32 \cdot 10^{-28}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 24.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -4.4 \cdot 10^{-26}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq 170000000000:\\ \;\;\;\;\frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -4.4e-26)
   (* NaChar 0.5)
   (if (<= NaChar 170000000000.0) (/ NdChar 2.0) (* NaChar 0.5))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -4.4e-26) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 170000000000.0) {
		tmp = NdChar / 2.0;
	} else {
		tmp = NaChar * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-4.4d-26)) then
        tmp = nachar * 0.5d0
    else if (nachar <= 170000000000.0d0) then
        tmp = ndchar / 2.0d0
    else
        tmp = nachar * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -4.4e-26) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 170000000000.0) {
		tmp = NdChar / 2.0;
	} else {
		tmp = NaChar * 0.5;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -4.4e-26:
		tmp = NaChar * 0.5
	elif NaChar <= 170000000000.0:
		tmp = NdChar / 2.0
	else:
		tmp = NaChar * 0.5
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -4.4e-26)
		tmp = Float64(NaChar * 0.5);
	elseif (NaChar <= 170000000000.0)
		tmp = Float64(NdChar / 2.0);
	else
		tmp = Float64(NaChar * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -4.4e-26)
		tmp = NaChar * 0.5;
	elseif (NaChar <= 170000000000.0)
		tmp = NdChar / 2.0;
	else
		tmp = NaChar * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -4.4e-26], N[(NaChar * 0.5), $MachinePrecision], If[LessEqual[NaChar, 170000000000.0], N[(NdChar / 2.0), $MachinePrecision], N[(NaChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -4.4000000000000002e-26 or 1.7e11 < NaChar

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 27.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 27.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   8. Taylor expanded in NaChar around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 25.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{NaChar}\right) \]

   10. Simplified 25.4%

       \[\leadsto \color{blue}{0.5 \cdot NaChar} \]

   if -4.4000000000000002e-26 < NaChar < 1.7e11

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EDonor + \left(Vef + mu\right)\right), Ec\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EDonor + Vef\right) + mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EDonor + Vef\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(Vef + EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

      9. +-lowering-+.f64 71.7%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), mu\right), Ec\right), KbT\right)\right)\right)\right) \]

   7. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}}} \]

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{2}\right) \]
   9. Step-by-step derivation

   10. Simplified 24.8%

       \[\leadsto \frac{NdChar}{\color{blue}{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -4.4 \cdot 10^{-26}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq 170000000000:\\ \;\;\;\;\frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 28.0% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -1.7 \cdot 10^{+210}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -1.7e+210)
   (/ NaChar (+ 2.0 (/ Ev KbT)))
   (* 0.5 (+ NdChar NaChar))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -1.7e+210) {
		tmp = NaChar / (2.0 + (Ev / KbT));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ev <= (-1.7d+210)) then
        tmp = nachar / (2.0d0 + (ev / kbt))
    else
        tmp = 0.5d0 * (ndchar + nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -1.7e+210) {
		tmp = NaChar / (2.0 + (Ev / KbT));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -1.7e+210:
		tmp = NaChar / (2.0 + (Ev / KbT))
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -1.7e+210)
		tmp = Float64(NaChar / Float64(2.0 + Float64(Ev / KbT)));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -1.7e+210)
		tmp = NaChar / (2.0 + (Ev / KbT));
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -1.7e+210], N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Ev < -1.70000000000000012e210

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}{NaChar}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot \color{blue}{NaChar}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}\right), \color{blue}{NaChar}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right), NaChar\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}\right)\right)\right), NaChar\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}\right)\right)\right)\right), NaChar\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \left(EAccept + \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \left(Ev - mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right), NaChar\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \color{blue}{\frac{1}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \cdot NaChar} \]

   7. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 63.7%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

   9. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   10. Taylor expanded in Ev around inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 58.5%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right)\right)\right)\right) \]

   12. Simplified 58.5%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   13. Taylor expanded in Ev around 0

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + \frac{Ev}{KbT}\right)}\right) \]
   14. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right) \]

       2. /-lowering-/.f64 22.4%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(Ev, \color{blue}{KbT}\right)\right)\right) \]

   15. Simplified 22.4%

       \[\leadsto \frac{NaChar}{\color{blue}{2 + \frac{Ev}{KbT}}} \]

   if -1.70000000000000012e210 < 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%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 28.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 28.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -1.7 \cdot 10^{+210}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 28.5% accurate, 45.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(NdChar + NaChar\right) \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* 0.5 (+ NdChar NaChar)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return 0.5 * (NdChar + NaChar);
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = 0.5d0 * (ndchar + nachar)
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return 0.5 * (NdChar + NaChar);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return 0.5 * (NdChar + NaChar)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(0.5 * Float64(NdChar + NaChar))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.5 * (NdChar + NaChar);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
4. Add Preprocessing

5. Taylor expanded in KbT around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
6. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

   3. +-lowering-+.f64 28.0%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

7. Simplified 28.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

8. Final simplification 28.0%

   \[\leadsto 0.5 \cdot \left(NdChar + NaChar\right) \]
9. Add Preprocessing

Alternative 31: 18.4% accurate, 76.3× speedup

Math

\[\begin{array}{l} \\ NaChar \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* NaChar 0.5))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NaChar * 0.5;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = nachar * 0.5d0
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NaChar * 0.5;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return NaChar * 0.5

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(NaChar * 0.5)
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = NaChar * 0.5;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(NaChar * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
4. Add Preprocessing

5. Taylor expanded in KbT around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
6. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

   3. +-lowering-+.f64 28.0%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

7. Simplified 28.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

8. Taylor expanded in NaChar around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 19.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{NaChar}\right) \]

10. Simplified 19.3%

    \[\leadsto \color{blue}{0.5 \cdot NaChar} \]

11. Final simplification 19.3%

    \[\leadsto NaChar \cdot 0.5 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(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))))))