Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 19.5s

Alternatives: 21

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 3.175921904837051e+210
  2. Ec = -3.869580975418154e-278
  3. Vef = 1.3627602683999797e+264
  4. EDonor = 5.03915919573418e+56
  5. mu = 2.5359467360461487e+212
  6. KbT = -2.616337373602224e-206
  7. NaChar = -3.391674021198336e+209
  8. Ev = 1.846046327619081e+133
  9. EAccept = 3.2351216271339324e-143

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (Math.pow(Math.E, ((Vef - (Ec - (mu + EDonor))) / 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.pow(math.e, ((Vef - (Ec - (mu + EDonor))) / 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(1) ^ Float64(Float64(Vef - Float64(Ec - Float64(mu + EDonor))) / 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 / ((2.71828182845904523536 ^ ((Vef - (Ec - (mu + EDonor))) / 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[Power[E, N[(N[(Vef - N[(Ec - N[(mu + EDonor), $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. Step-by-step derivation

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. clear-num N/A

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

   4. exp-prod N/A

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

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

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

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

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

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

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

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

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

   9. associate-+r- N/A

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

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

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

   11. +-lowering-+.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} - \frac{NdChar}{-1 - e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}}\\ \mathbf{if}\;Vef \leq -15000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
          (/ NdChar (- -1.0 (exp (/ (+ Vef (+ mu (- EDonor Ec))) KbT)))))))
   (if (<= Vef -15000000000000.0)
     t_0
     (if (<= Vef 1.35e-76)
       (+
        (/ NaChar (+ (exp (/ (+ Vef (+ EAccept (- Ev mu))) KbT)) 1.0))
        (/ NdChar (+ (exp (/ EDonor 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 = (NaChar / (exp((Vef / KbT)) + 1.0)) - (NdChar / (-1.0 - exp(((Vef + (mu + (EDonor - Ec))) / KbT))));
	double tmp;
	if (Vef <= -15000000000000.0) {
		tmp = t_0;
	} else if (Vef <= 1.35e-76) {
		tmp = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (exp((EDonor / 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 = (nachar / (exp((vef / kbt)) + 1.0d0)) - (ndchar / ((-1.0d0) - exp(((vef + (mu + (edonor - ec))) / kbt))))
    if (vef <= (-15000000000000.0d0)) then
        tmp = t_0
    else if (vef <= 1.35d-76) then
        tmp = (nachar / (exp(((vef + (eaccept + (ev - mu))) / kbt)) + 1.0d0)) + (ndchar / (exp((edonor / 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 = (NaChar / (Math.exp((Vef / KbT)) + 1.0)) - (NdChar / (-1.0 - Math.exp(((Vef + (mu + (EDonor - Ec))) / KbT))));
	double tmp;
	if (Vef <= -15000000000000.0) {
		tmp = t_0;
	} else if (Vef <= 1.35e-76) {
		tmp = (NaChar / (Math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (Math.exp((EDonor / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (math.exp((Vef / KbT)) + 1.0)) - (NdChar / (-1.0 - math.exp(((Vef + (mu + (EDonor - Ec))) / KbT))))
	tmp = 0
	if Vef <= -15000000000000.0:
		tmp = t_0
	elif Vef <= 1.35e-76:
		tmp = (NaChar / (math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (math.exp((EDonor / 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(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(Vef + Float64(mu + Float64(EDonor - Ec))) / KbT)))))
	tmp = 0.0
	if (Vef <= -15000000000000.0)
		tmp = t_0;
	elseif (Vef <= 1.35e-76)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(EAccept + Float64(Ev - mu))) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(EDonor / 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 = (NaChar / (exp((Vef / KbT)) + 1.0)) - (NdChar / (-1.0 - exp(((Vef + (mu + (EDonor - Ec))) / KbT))));
	tmp = 0.0;
	if (Vef <= -15000000000000.0)
		tmp = t_0;
	elseif (Vef <= 1.35e-76)
		tmp = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (exp((EDonor / 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[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -15000000000000.0], t$95$0, If[LessEqual[Vef, 1.35e-76], 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 / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -1.5e13 or 1.35e-76 < Vef

   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 83.4%

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

   7. Simplified 83.4%

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

   if -1.5e13 < Vef < 1.35e-76

   1. Initial program 100.0%

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

   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 EDonor around inf

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

      1. /-lowering-/.f64 78.2%

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

   7. Simplified 78.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -15000000000000:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} - \frac{NdChar}{-1 - e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} - \frac{NdChar}{-1 - e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NaChar (+ (exp (/ (+ Vef (+ EAccept (- Ev mu))) KbT)) 1.0))
  (/ NdChar (+ (exp (/ (+ Vef (+ mu (- EDonor Ec))) 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 (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / 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 = (nachar / (exp(((vef + (eaccept + (ev - mu))) / kbt)) + 1.0d0)) + (ndchar / (exp(((vef + (mu + (edonor - ec))) / 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 (NaChar / (Math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (Math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 4: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;mu \leq 1.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= mu 1.6e+91)
   (+
    (/ NaChar (+ (exp (/ (+ Vef (+ EAccept (- Ev mu))) KbT)) 1.0))
    (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)))
   (/ NdChar (+ (exp (/ (- (+ mu (+ Vef EDonor)) Ec) 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 <= 1.6e+91) {
		tmp = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (exp((EDonor / KbT)) + 1.0));
	} else {
		tmp = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / 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 <= 1.6d+91) then
        tmp = (nachar / (exp(((vef + (eaccept + (ev - mu))) / kbt)) + 1.0d0)) + (ndchar / (exp((edonor / kbt)) + 1.0d0))
    else
        tmp = ndchar / (exp((((mu + (vef + edonor)) - ec) / 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 <= 1.6e+91) {
		tmp = (NaChar / (Math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (Math.exp((EDonor / KbT)) + 1.0));
	} else {
		tmp = NdChar / (Math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if mu <= 1.6e+91:
		tmp = (NaChar / (math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (math.exp((EDonor / KbT)) + 1.0))
	else:
		tmp = NdChar / (math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (mu <= 1.6e+91)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(EAccept + Float64(Ev - mu))) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + Float64(Vef + EDonor)) - Ec) / 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 <= 1.6e+91)
		tmp = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (exp((EDonor / KbT)) + 1.0));
	else
		tmp = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[mu, 1.6e+91], 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 / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(N[(N[(mu + N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if mu < 1.59999999999999995e91

   1. Initial program 100.0%

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

   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 EDonor around inf

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

      1. /-lowering-/.f64 71.8%

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

   7. Simplified 71.8%

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

   if 1.59999999999999995e91 < mu

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq 1.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{if}\;KbT \leq -5.8 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq -1.2 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq -4 \cdot 10^{-210}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 8.5 \cdot 10^{-186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))
        (t_1 (+ (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (/ NaChar 2.0))))
   (if (<= KbT -5.8e+144)
     t_1
     (if (<= KbT -1.2e-120)
       t_0
       (if (<= KbT -4e-210)
         (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
         (if (<= KbT 8.5e-186)
           t_0
           (if (<= KbT 1.05e+139)
             (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
             t_1)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((EAccept / KbT)) + 1.0);
	double t_1 = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	double tmp;
	if (KbT <= -5.8e+144) {
		tmp = t_1;
	} else if (KbT <= -1.2e-120) {
		tmp = t_0;
	} else if (KbT <= -4e-210) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else if (KbT <= 8.5e-186) {
		tmp = t_0;
	} else if (KbT <= 1.05e+139) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (exp((eaccept / kbt)) + 1.0d0)
    t_1 = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar / 2.0d0)
    if (kbt <= (-5.8d+144)) then
        tmp = t_1
    else if (kbt <= (-1.2d-120)) then
        tmp = t_0
    else if (kbt <= (-4d-210)) then
        tmp = ndchar / (exp((vef / kbt)) + 1.0d0)
    else if (kbt <= 8.5d-186) then
        tmp = t_0
    else if (kbt <= 1.05d+139) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	double t_1 = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	double tmp;
	if (KbT <= -5.8e+144) {
		tmp = t_1;
	} else if (KbT <= -1.2e-120) {
		tmp = t_0;
	} else if (KbT <= -4e-210) {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	} else if (KbT <= 8.5e-186) {
		tmp = t_0;
	} else if (KbT <= 1.05e+139) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	t_1 = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0)
	tmp = 0
	if KbT <= -5.8e+144:
		tmp = t_1
	elif KbT <= -1.2e-120:
		tmp = t_0
	elif KbT <= -4e-210:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	elif KbT <= 8.5e-186:
		tmp = t_0
	elif KbT <= 1.05e+139:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar / 2.0))
	tmp = 0.0
	if (KbT <= -5.8e+144)
		tmp = t_1;
	elseif (KbT <= -1.2e-120)
		tmp = t_0;
	elseif (KbT <= -4e-210)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (KbT <= 8.5e-186)
		tmp = t_0;
	elseif (KbT <= 1.05e+139)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((EAccept / KbT)) + 1.0);
	t_1 = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	tmp = 0.0;
	if (KbT <= -5.8e+144)
		tmp = t_1;
	elseif (KbT <= -1.2e-120)
		tmp = t_0;
	elseif (KbT <= -4e-210)
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	elseif (KbT <= 8.5e-186)
		tmp = t_0;
	elseif (KbT <= 1.05e+139)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -5.8e+144], t$95$1, If[LessEqual[KbT, -1.2e-120], t$95$0, If[LessEqual[KbT, -4e-210], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 8.5e-186], t$95$0, If[LessEqual[KbT, 1.05e+139], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -5.79999999999999996e144 or 1.0499999999999999e139 < 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 EDonor around inf

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

      1. /-lowering-/.f64 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in KbT around inf

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

   10. Simplified 77.1%

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

   if -5.79999999999999996e144 < KbT < -1.2e-120 or -4.0000000000000002e-210 < KbT < 8.4999999999999994e-186

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 66.0%

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

   7. Simplified 66.0%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 35.8%

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

   10. Simplified 35.8%

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

   if -1.2e-120 < KbT < -4.0000000000000002e-210

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 87.6%

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

   7. Simplified 87.6%

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

   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 56.1%

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

   10. Simplified 56.1%

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

   if 8.4999999999999994e-186 < KbT < 1.0499999999999999e139

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in Ev around inf

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

      1. /-lowering-/.f64 43.5%

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

   10. Simplified 43.5%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq -1.2 \cdot 10^{-120}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq -4 \cdot 10^{-210}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 8.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ t_1 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;KbT \leq -1.1 \cdot 10^{+139}:\\ \;\;\;\;t\_0 + \frac{NdChar}{2}\\ \mathbf{elif}\;KbT \leq -3.7 \cdot 10^{-121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq -8.4 \cdot 10^{-211}:\\ \;\;\;\;\frac{NdChar}{t\_1}\\ \mathbf{elif}\;KbT \leq 2 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 4.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_1} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))
        (t_1 (+ (exp (/ Vef KbT)) 1.0)))
   (if (<= KbT -1.1e+139)
     (+ t_0 (/ NdChar 2.0))
     (if (<= KbT -3.7e-121)
       t_0
       (if (<= KbT -8.4e-211)
         (/ NdChar t_1)
         (if (<= KbT 2e-188)
           t_0
           (if (<= KbT 4.8e+46)
             (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
             (+ (/ NaChar t_1) (/ NdChar 2.0)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((EAccept / KbT)) + 1.0);
	double t_1 = exp((Vef / KbT)) + 1.0;
	double tmp;
	if (KbT <= -1.1e+139) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (KbT <= -3.7e-121) {
		tmp = t_0;
	} else if (KbT <= -8.4e-211) {
		tmp = NdChar / t_1;
	} else if (KbT <= 2e-188) {
		tmp = t_0;
	} else if (KbT <= 4.8e+46) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else {
		tmp = (NaChar / t_1) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (exp((eaccept / kbt)) + 1.0d0)
    t_1 = exp((vef / kbt)) + 1.0d0
    if (kbt <= (-1.1d+139)) then
        tmp = t_0 + (ndchar / 2.0d0)
    else if (kbt <= (-3.7d-121)) then
        tmp = t_0
    else if (kbt <= (-8.4d-211)) then
        tmp = ndchar / t_1
    else if (kbt <= 2d-188) then
        tmp = t_0
    else if (kbt <= 4.8d+46) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else
        tmp = (nachar / t_1) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	double t_1 = Math.exp((Vef / KbT)) + 1.0;
	double tmp;
	if (KbT <= -1.1e+139) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (KbT <= -3.7e-121) {
		tmp = t_0;
	} else if (KbT <= -8.4e-211) {
		tmp = NdChar / t_1;
	} else if (KbT <= 2e-188) {
		tmp = t_0;
	} else if (KbT <= 4.8e+46) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else {
		tmp = (NaChar / t_1) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	t_1 = math.exp((Vef / KbT)) + 1.0
	tmp = 0
	if KbT <= -1.1e+139:
		tmp = t_0 + (NdChar / 2.0)
	elif KbT <= -3.7e-121:
		tmp = t_0
	elif KbT <= -8.4e-211:
		tmp = NdChar / t_1
	elif KbT <= 2e-188:
		tmp = t_0
	elif KbT <= 4.8e+46:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	else:
		tmp = (NaChar / t_1) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0))
	t_1 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	tmp = 0.0
	if (KbT <= -1.1e+139)
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	elseif (KbT <= -3.7e-121)
		tmp = t_0;
	elseif (KbT <= -8.4e-211)
		tmp = Float64(NdChar / t_1);
	elseif (KbT <= 2e-188)
		tmp = t_0;
	elseif (KbT <= 4.8e+46)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	else
		tmp = Float64(Float64(NaChar / t_1) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((EAccept / KbT)) + 1.0);
	t_1 = exp((Vef / KbT)) + 1.0;
	tmp = 0.0;
	if (KbT <= -1.1e+139)
		tmp = t_0 + (NdChar / 2.0);
	elseif (KbT <= -3.7e-121)
		tmp = t_0;
	elseif (KbT <= -8.4e-211)
		tmp = NdChar / t_1;
	elseif (KbT <= 2e-188)
		tmp = t_0;
	elseif (KbT <= 4.8e+46)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	else
		tmp = (NaChar / t_1) + (NdChar / 2.0);
	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[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[KbT, -1.1e+139], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -3.7e-121], t$95$0, If[LessEqual[KbT, -8.4e-211], N[(NdChar / t$95$1), $MachinePrecision], If[LessEqual[KbT, 2e-188], t$95$0, If[LessEqual[KbT, 4.8e+46], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / t$95$1), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if KbT < -1.1e139

   1. Initial program 100.0%

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

   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

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

   7. Simplified 82.8%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 82.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right)\right)\right)\right)\right) \]

   10. Simplified 82.4%

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

   if -1.1e139 < KbT < -3.7000000000000002e-121 or -8.4000000000000003e-211 < KbT < 1.9999999999999999e-188

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 66.0%

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

   7. Simplified 66.0%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 35.8%

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

   10. Simplified 35.8%

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

   if -3.7000000000000002e-121 < KbT < -8.4000000000000003e-211

   1. Initial program 100.0%

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

   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 87.6%

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

   7. Simplified 87.6%

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

   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 56.1%

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

   10. Simplified 56.1%

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

   if 1.9999999999999999e-188 < KbT < 4.80000000000000017e46

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 61.3%

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

   7. Simplified 61.3%

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

   8. Taylor expanded in Ev around inf

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

      1. /-lowering-/.f64 41.2%

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

   10. Simplified 41.2%

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

   if 4.80000000000000017e46 < KbT

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 66.2%

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

   8. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 58.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right)\right) \]

   10. Simplified 58.7%

       \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + \frac{NdChar}{2}\\ \mathbf{elif}\;KbT \leq -3.7 \cdot 10^{-121}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq -8.4 \cdot 10^{-211}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 2 \cdot 10^{-188}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 4.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{if}\;EDonor \leq -1.72 \cdot 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;EDonor \leq -1.85 \cdot 10^{-155}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 2.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 6.2 \cdot 10^{+235}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))))
   (if (<= EDonor -1.72e+115)
     t_0
     (if (<= EDonor -1.85e-155)
       (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
       (if (<= EDonor 2.6e+116)
         (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
         (if (<= EDonor 6.2e+235) (/ NaChar (+ (exp (/ Ev KbT)) 1.0)) t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double tmp;
	if (EDonor <= -1.72e+115) {
		tmp = t_0;
	} else if (EDonor <= -1.85e-155) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (EDonor <= 2.6e+116) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else if (EDonor <= 6.2e+235) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp((edonor / kbt)) + 1.0d0)
    if (edonor <= (-1.72d+115)) then
        tmp = t_0
    else if (edonor <= (-1.85d-155)) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else if (edonor <= 2.6d+116) then
        tmp = ndchar / (exp((vef / kbt)) + 1.0d0)
    else if (edonor <= 6.2d+235) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	double tmp;
	if (EDonor <= -1.72e+115) {
		tmp = t_0;
	} else if (EDonor <= -1.85e-155) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else if (EDonor <= 2.6e+116) {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	} else if (EDonor <= 6.2e+235) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	tmp = 0
	if EDonor <= -1.72e+115:
		tmp = t_0
	elif EDonor <= -1.85e-155:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	elif EDonor <= 2.6e+116:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	elif EDonor <= 6.2e+235:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	tmp = 0.0
	if (EDonor <= -1.72e+115)
		tmp = t_0;
	elseif (EDonor <= -1.85e-155)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (EDonor <= 2.6e+116)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (EDonor <= 6.2e+235)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((EDonor / KbT)) + 1.0);
	tmp = 0.0;
	if (EDonor <= -1.72e+115)
		tmp = t_0;
	elseif (EDonor <= -1.85e-155)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	elseif (EDonor <= 2.6e+116)
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	elseif (EDonor <= 6.2e+235)
		tmp = NaChar / (exp((Ev / 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[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EDonor, -1.72e+115], t$95$0, If[LessEqual[EDonor, -1.85e-155], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 2.6e+116], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 6.2e+235], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if EDonor < -1.72e115 or 6.20000000000000022e235 < EDonor

   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 EDonor around inf

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

      1. /-lowering-/.f64 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 57.3%

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

   10. Simplified 57.3%

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

   if -1.72e115 < EDonor < -1.85e-155

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 69.0%

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

   7. Simplified 69.0%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 51.3%

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

   10. Simplified 51.3%

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

   if -1.85e-155 < EDonor < 2.59999999999999987e116

   1. Initial program 100.0%

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

   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 66.3%

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

   7. Simplified 66.3%

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

   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 52.1%

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

   10. Simplified 52.1%

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

   if 2.59999999999999987e116 < EDonor < 6.20000000000000022e235

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in Ev around inf

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

      1. /-lowering-/.f64 66.1%

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

   10. Simplified 66.1%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -1.72 \cdot 10^{+115}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq -1.85 \cdot 10^{-155}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 2.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 6.2 \cdot 10^{+235}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 2.85 \cdot 10^{-27}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))))
   (if (<= NaChar -5.8e-6)
     t_0
     (if (<= NaChar 2.85e-27)
       (/ NdChar (+ (exp (/ (- (+ mu (+ Vef EDonor)) Ec) 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 = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -5.8e-6) {
		tmp = t_0;
	} else if (NaChar <= 2.85e-27) {
		tmp = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / 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 = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    if (nachar <= (-5.8d-6)) then
        tmp = t_0
    else if (nachar <= 2.85d-27) then
        tmp = ndchar / (exp((((mu + (vef + edonor)) - ec) / 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 = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -5.8e-6) {
		tmp = t_0;
	} else if (NaChar <= 2.85e-27) {
		tmp = NdChar / (Math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	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)
	tmp = 0
	if NaChar <= -5.8e-6:
		tmp = t_0
	elif NaChar <= 2.85e-27:
		tmp = NdChar / (math.exp((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -5.8e-6)
		tmp = t_0;
	elseif (NaChar <= 2.85e-27)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + Float64(Vef + EDonor)) - Ec) / 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 = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -5.8e-6)
		tmp = t_0;
	elseif (NaChar <= 2.85e-27)
		tmp = NdChar / (exp((((mu + (Vef + EDonor)) - Ec) / 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[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -5.8e-6], t$95$0, If[LessEqual[NaChar, 2.85e-27], N[(NdChar / N[(N[Exp[N[(N[(N[(mu + N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -5.8000000000000004e-6 or 2.8499999999999998e-27 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 72.0%

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

   7. Simplified 72.0%

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

   if -5.8000000000000004e-6 < NaChar < 2.8499999999999998e-27

   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 72.1%

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

   7. Simplified 72.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 2.85 \cdot 10^{-27}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{if}\;KbT \leq -2 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 10^{+249}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (/ NaChar 2.0))))
   (if (<= KbT -2e+141)
     t_0
     (if (<= KbT 1e+249)
       (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	double tmp;
	if (KbT <= -2e+141) {
		tmp = t_0;
	} else if (KbT <= 1e+249) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar / 2.0d0)
    if (kbt <= (-2d+141)) then
        tmp = t_0
    else if (kbt <= 1d+249) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	double tmp;
	if (KbT <= -2e+141) {
		tmp = t_0;
	} else if (KbT <= 1e+249) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0)
	tmp = 0
	if KbT <= -2e+141:
		tmp = t_0
	elif KbT <= 1e+249:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar / 2.0))
	tmp = 0.0
	if (KbT <= -2e+141)
		tmp = t_0;
	elseif (KbT <= 1e+249)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	tmp = 0.0;
	if (KbT <= -2e+141)
		tmp = t_0;
	elseif (KbT <= 1e+249)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -2e+141], t$95$0, If[LessEqual[KbT, 1e+249], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -2.00000000000000003e141 or 9.9999999999999992e248 < 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 EDonor around inf

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

      1. /-lowering-/.f64 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in KbT around inf

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

   10. Simplified 88.1%

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

   if -2.00000000000000003e141 < KbT < 9.9999999999999992e248

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 62.5%

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

   7. Simplified 62.5%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 10^{+249}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -1.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.45 \cdot 10^{-186}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 4.2 \cdot 10^{+139}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -1.9e+136)
   (+
    (/ NdChar 2.0)
    (/
     NaChar
     (- (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Vef KbT) (/ Ev KbT))) (/ mu KbT))))
   (if (<= KbT 1.45e-186)
     (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
     (if (<= KbT 4.2e+139)
       (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
       (* 0.5 (+ NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.9e+136) {
		tmp = (NdChar / 2.0) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else if (KbT <= 1.45e-186) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (KbT <= 4.2e+139) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-1.9d+136)) then
        tmp = (ndchar / 2.0d0) + (nachar / (((2.0d0 + (eaccept / kbt)) + ((vef / kbt) + (ev / kbt))) - (mu / kbt)))
    else if (kbt <= 1.45d-186) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else if (kbt <= 4.2d+139) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else
        tmp = 0.5d0 * (ndchar + nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.9e+136) {
		tmp = (NdChar / 2.0) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else if (KbT <= 1.45e-186) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else if (KbT <= 4.2e+139) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -1.9e+136:
		tmp = (NdChar / 2.0) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)))
	elif KbT <= 1.45e-186:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	elif KbT <= 4.2e+139:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -1.9e+136)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) - Float64(mu / KbT))));
	elseif (KbT <= 1.45e-186)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (KbT <= 4.2e+139)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -1.9e+136)
		tmp = (NdChar / 2.0) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	elseif (KbT <= 1.45e-186)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	elseif (KbT <= 4.2e+139)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -1.9e+136], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.45e-186], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.2e+139], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -1.90000000000000007e136

   1. Initial program 100.0%

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

   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

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

   7. Simplified 81.3%

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

   8. Taylor expanded in KbT around inf

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

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

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

      2. associate-+r+ N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EAccept}{KbT}\right)\right), \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \left(\frac{Ev}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \left(\frac{Ev}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 79.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right)\right), \mathsf{/.f64}\left(mu, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 79.0%

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

   if -1.90000000000000007e136 < KbT < 1.4500000000000001e-186

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 34.0%

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

   10. Simplified 34.0%

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

   if 1.4500000000000001e-186 < KbT < 4.1999999999999997e139

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in Ev around inf

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

      1. /-lowering-/.f64 43.5%

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

   10. Simplified 43.5%

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

   if 4.1999999999999997e139 < KbT

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
   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 55.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 55.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.45 \cdot 10^{-186}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 4.2 \cdot 10^{+139}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -1.55 \cdot 10^{+136}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 2.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -1.55e+136)
   (+
    (/ NdChar 2.0)
    (/
     NaChar
     (- (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Vef KbT) (/ Ev KbT))) (/ mu KbT))))
   (if (<= KbT 2.9e+27)
     (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
     (* 0.5 (+ NdChar NaChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.55e+136) {
		tmp = (NdChar / 2.0) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else if (KbT <= 2.9e+27) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-1.55d+136)) then
        tmp = (ndchar / 2.0d0) + (nachar / (((2.0d0 + (eaccept / kbt)) + ((vef / kbt) + (ev / kbt))) - (mu / kbt)))
    else if (kbt <= 2.9d+27) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else
        tmp = 0.5d0 * (ndchar + nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.55e+136) {
		tmp = (NdChar / 2.0) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else if (KbT <= 2.9e+27) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -1.55e+136:
		tmp = (NdChar / 2.0) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)))
	elif KbT <= 2.9e+27:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -1.55e+136)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) - Float64(mu / KbT))));
	elseif (KbT <= 2.9e+27)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -1.55e+136)
		tmp = (NdChar / 2.0) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	elseif (KbT <= 2.9e+27)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -1.55e+136], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.9e+27], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.54999999999999992e136

   1. Initial program 100.0%

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

   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

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

   7. Simplified 81.3%

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

   8. Taylor expanded in KbT around inf

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

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

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

      2. associate-+r+ N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EAccept}{KbT}\right)\right), \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \left(\frac{Ev}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \left(\frac{Ev}{KbT}\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right)\right), \left(\frac{mu}{KbT}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 79.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, KbT\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), \mathsf{/.f64}\left(Ev, KbT\right)\right)\right), \mathsf{/.f64}\left(mu, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 79.0%

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

   if -1.54999999999999992e136 < KbT < 2.9000000000000001e27

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 62.2%

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

   7. Simplified 62.2%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 37.0%

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

   10. Simplified 37.0%

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

   if 2.9000000000000001e27 < 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 46.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 46.7%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.55 \cdot 10^{+136}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 2.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -4.2 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 3.7 \cdot 10^{-61}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) + \frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT} \cdot \left(\left(\left(\left(Ec - EDonor\right) - mu\right) - Vef\right) \cdot -0.5\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -4.2e+83)
     t_0
     (if (<= KbT 3.7e-61)
       (/
        NdChar
        (+
         2.0
         (/
          (+
           (+ (+ Vef EDonor) (- mu Ec))
           (*
            (/ (+ Vef (+ mu (- EDonor Ec))) KbT)
            (* (- (- (- Ec EDonor) mu) Vef) -0.5)))
          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 <= -4.2e+83) {
		tmp = t_0;
	} else if (KbT <= 3.7e-61) {
		tmp = NdChar / (2.0 + ((((Vef + EDonor) + (mu - Ec)) + (((Vef + (mu + (EDonor - Ec))) / KbT) * ((((Ec - EDonor) - mu) - Vef) * -0.5))) / 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 <= (-4.2d+83)) then
        tmp = t_0
    else if (kbt <= 3.7d-61) then
        tmp = ndchar / (2.0d0 + ((((vef + edonor) + (mu - ec)) + (((vef + (mu + (edonor - ec))) / kbt) * ((((ec - edonor) - mu) - vef) * (-0.5d0)))) / 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 <= -4.2e+83) {
		tmp = t_0;
	} else if (KbT <= 3.7e-61) {
		tmp = NdChar / (2.0 + ((((Vef + EDonor) + (mu - Ec)) + (((Vef + (mu + (EDonor - Ec))) / KbT) * ((((Ec - EDonor) - mu) - Vef) * -0.5))) / 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 <= -4.2e+83:
		tmp = t_0
	elif KbT <= 3.7e-61:
		tmp = NdChar / (2.0 + ((((Vef + EDonor) + (mu - Ec)) + (((Vef + (mu + (EDonor - Ec))) / KbT) * ((((Ec - EDonor) - mu) - Vef) * -0.5))) / 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 <= -4.2e+83)
		tmp = t_0;
	elseif (KbT <= 3.7e-61)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Float64(Vef + EDonor) + Float64(mu - Ec)) + Float64(Float64(Float64(Vef + Float64(mu + Float64(EDonor - Ec))) / KbT) * Float64(Float64(Float64(Float64(Ec - EDonor) - mu) - Vef) * -0.5))) / 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 <= -4.2e+83)
		tmp = t_0;
	elseif (KbT <= 3.7e-61)
		tmp = NdChar / (2.0 + ((((Vef + EDonor) + (mu - Ec)) + (((Vef + (mu + (EDonor - Ec))) / KbT) * ((((Ec - EDonor) - mu) - Vef) * -0.5))) / 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, -4.2e+83], t$95$0, If[LessEqual[KbT, 3.7e-61], N[(NdChar / N[(2.0 + N[(N[(N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision] * N[(N[(N[(N[(Ec - EDonor), $MachinePrecision] - mu), $MachinePrecision] - Vef), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -4.20000000000000005e83 or 3.7e-61 < 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 51.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 51.1%

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

   if -4.20000000000000005e83 < KbT < 3.7e-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 64.5%

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

   7. Simplified 64.5%

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

   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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   10. Simplified 28.2%

       \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\frac{-0.5 \cdot \left(\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right)\right)}{KbT} - \left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right)}{KbT}}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

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

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

       4. *-commutative N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right), \frac{-1}{2}\right), \left(\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

       6. associate-+r- N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(Vef + EDonor\right) + mu\right) - Ec\right), \frac{-1}{2}\right), \left(\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

       7. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(Vef + \left(EDonor + mu\right)\right) - Ec\right), \frac{-1}{2}\right), \left(\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(Vef + \left(mu + EDonor\right)\right) - Ec\right), \frac{-1}{2}\right), \left(\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

       9. associate-+r- N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right), \frac{-1}{2}\right), \left(\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

       10. associate-+r- N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right), \frac{-1}{2}\right), \left(\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right), \frac{-1}{2}\right), \left(\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \left(EDonor - Ec\right)\right)\right), \frac{-1}{2}\right), \left(\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

       13. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), \frac{-1}{2}\right), \left(\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right), KbT\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(Vef, EDonor\right), \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right) \]

   12. Applied egg-rr 28.2%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.2 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 3.7 \cdot 10^{-61}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) + \frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT} \cdot \left(\left(\left(\left(Ec - EDonor\right) - mu\right) - Vef\right) \cdot -0.5\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.75 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.4 \cdot 10^{-197}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(Vef + mu\right) + \left(EDonor - Ec\right)}{KbT}}\\ \mathbf{elif}\;KbT \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{\frac{EAccept \cdot EAccept}{KbT} \cdot -0.5 - EAccept}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.75e+79)
     t_0
     (if (<= KbT 2.4e-197)
       (/ NdChar (+ 2.0 (/ (+ (+ Vef mu) (- EDonor Ec)) KbT)))
       (if (<= KbT 8.5e-17)
         (/
          NaChar
          (- 2.0 (/ (- (* (/ (* EAccept EAccept) KbT) -0.5) EAccept) 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 <= -1.75e+79) {
		tmp = t_0;
	} else if (KbT <= 2.4e-197) {
		tmp = NdChar / (2.0 + (((Vef + mu) + (EDonor - Ec)) / KbT));
	} else if (KbT <= 8.5e-17) {
		tmp = NaChar / (2.0 - (((((EAccept * EAccept) / KbT) * -0.5) - EAccept) / 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 <= (-1.75d+79)) then
        tmp = t_0
    else if (kbt <= 2.4d-197) then
        tmp = ndchar / (2.0d0 + (((vef + mu) + (edonor - ec)) / kbt))
    else if (kbt <= 8.5d-17) then
        tmp = nachar / (2.0d0 - (((((eaccept * eaccept) / kbt) * (-0.5d0)) - eaccept) / 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 <= -1.75e+79) {
		tmp = t_0;
	} else if (KbT <= 2.4e-197) {
		tmp = NdChar / (2.0 + (((Vef + mu) + (EDonor - Ec)) / KbT));
	} else if (KbT <= 8.5e-17) {
		tmp = NaChar / (2.0 - (((((EAccept * EAccept) / KbT) * -0.5) - EAccept) / 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 <= -1.75e+79:
		tmp = t_0
	elif KbT <= 2.4e-197:
		tmp = NdChar / (2.0 + (((Vef + mu) + (EDonor - Ec)) / KbT))
	elif KbT <= 8.5e-17:
		tmp = NaChar / (2.0 - (((((EAccept * EAccept) / KbT) * -0.5) - EAccept) / 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 <= -1.75e+79)
		tmp = t_0;
	elseif (KbT <= 2.4e-197)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Vef + mu) + Float64(EDonor - Ec)) / KbT)));
	elseif (KbT <= 8.5e-17)
		tmp = Float64(NaChar / Float64(2.0 - Float64(Float64(Float64(Float64(Float64(EAccept * EAccept) / KbT) * -0.5) - EAccept) / 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 <= -1.75e+79)
		tmp = t_0;
	elseif (KbT <= 2.4e-197)
		tmp = NdChar / (2.0 + (((Vef + mu) + (EDonor - Ec)) / KbT));
	elseif (KbT <= 8.5e-17)
		tmp = NaChar / (2.0 - (((((EAccept * EAccept) / KbT) * -0.5) - EAccept) / 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, -1.75e+79], t$95$0, If[LessEqual[KbT, 2.4e-197], N[(NdChar / N[(2.0 + N[(N[(N[(Vef + mu), $MachinePrecision] + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 8.5e-17], N[(NaChar / N[(2.0 - N[(N[(N[(N[(N[(EAccept * EAccept), $MachinePrecision] / KbT), $MachinePrecision] * -0.5), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.7499999999999999e79 or 8.5e-17 < 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 54.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 54.0%

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

   if -1.7499999999999999e79 < KbT < 2.4000000000000001e-197

   1. Initial program 100.0%

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

   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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   10. Simplified 26.3%

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

   11. Taylor expanded in KbT around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right), \color{blue}{KbT}\right)\right)\right) \]

       2. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(\left(Ec - EDonor\right) - \left(Vef + mu\right)\right), KbT\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(Ec - EDonor\right), \left(Vef + mu\right)\right), KbT\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(Ec, EDonor\right), \left(Vef + mu\right)\right), KbT\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(Ec, EDonor\right), \left(mu + Vef\right)\right), KbT\right)\right)\right) \]

       6. +-lowering-+.f64 23.1%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(Ec, EDonor\right), \mathsf{+.f64}\left(mu, Vef\right)\right), KbT\right)\right)\right) \]

   13. Simplified 23.1%

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

   if 2.4000000000000001e-197 < KbT < 8.5e-17

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 44.5%

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

   10. Simplified 44.5%

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

   11. Taylor expanded in KbT around -inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot EAccept + \frac{-1}{2} \cdot \frac{{EAccept}^{2}}{KbT}}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \left(2 + \left(\mathsf{neg}\left(\frac{-1 \cdot EAccept + \frac{-1}{2} \cdot \frac{{EAccept}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \left(2 - \color{blue}{\frac{-1 \cdot EAccept + \frac{-1}{2} \cdot \frac{{EAccept}^{2}}{KbT}}{KbT}}\right)\right) \]

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

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

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

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

       5. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{{EAccept}^{2}}{KbT} + -1 \cdot EAccept\right), KbT\right)\right)\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{{EAccept}^{2}}{KbT} + \left(\mathsf{neg}\left(EAccept\right)\right)\right), KbT\right)\right)\right) \]

       7. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{{EAccept}^{2}}{KbT} - EAccept\right), KbT\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{2} \cdot \frac{{EAccept}^{2}}{KbT}\right), EAccept\right), KbT\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{{EAccept}^{2}}{KbT}\right)\right), EAccept\right), KbT\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left({EAccept}^{2}\right), KbT\right)\right), EAccept\right), KbT\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(EAccept \cdot EAccept\right), KbT\right)\right), EAccept\right), KbT\right)\right)\right) \]

       12. *-lowering-*.f64 19.0%

           \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(EAccept, EAccept\right), KbT\right)\right), EAccept\right), KbT\right)\right)\right) \]

   13. Simplified 19.0%

       \[\leadsto \frac{NaChar}{\color{blue}{2 - \frac{-0.5 \cdot \frac{EAccept \cdot EAccept}{KbT} - EAccept}{KbT}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.75 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.4 \cdot 10^{-197}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(Vef + mu\right) + \left(EDonor - Ec\right)}{KbT}}\\ \mathbf{elif}\;KbT \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{\frac{EAccept \cdot EAccept}{KbT} \cdot -0.5 - EAccept}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -8000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 8.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{NaChar}{2 + EAccept \cdot \left(\frac{1}{KbT} + 0.5 \cdot \frac{EAccept}{KbT \cdot KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -8000000000.0)
     t_0
     (if (<= KbT 8.2e-19)
       (/
        NaChar
        (+ 2.0 (* EAccept (+ (/ 1.0 KbT) (* 0.5 (/ EAccept (* 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -8000000000.0) {
		tmp = t_0;
	} else if (KbT <= 8.2e-19) {
		tmp = NaChar / (2.0 + (EAccept * ((1.0 / KbT) + (0.5 * (EAccept / (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) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-8000000000.0d0)) then
        tmp = t_0
    else if (kbt <= 8.2d-19) then
        tmp = nachar / (2.0d0 + (eaccept * ((1.0d0 / kbt) + (0.5d0 * (eaccept / (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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -8000000000.0) {
		tmp = t_0;
	} else if (KbT <= 8.2e-19) {
		tmp = NaChar / (2.0 + (EAccept * ((1.0 / KbT) + (0.5 * (EAccept / (KbT * 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 <= -8000000000.0:
		tmp = t_0
	elif KbT <= 8.2e-19:
		tmp = NaChar / (2.0 + (EAccept * ((1.0 / KbT) + (0.5 * (EAccept / (KbT * 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 <= -8000000000.0)
		tmp = t_0;
	elseif (KbT <= 8.2e-19)
		tmp = Float64(NaChar / Float64(2.0 + Float64(EAccept * Float64(Float64(1.0 / KbT) + Float64(0.5 * Float64(EAccept / Float64(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 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -8000000000.0)
		tmp = t_0;
	elseif (KbT <= 8.2e-19)
		tmp = NaChar / (2.0 + (EAccept * ((1.0 / KbT) + (0.5 * (EAccept / (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[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -8000000000.0], t$95$0, If[LessEqual[KbT, 8.2e-19], N[(NaChar / N[(2.0 + N[(EAccept * N[(N[(1.0 / KbT), $MachinePrecision] + N[(0.5 * N[(EAccept / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -8e9 or 8.1999999999999997e-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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   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 50.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 50.8%

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

   if -8e9 < 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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 38.2%

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

   10. Simplified 38.2%

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

   11. Taylor expanded in EAccept around 0

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + EAccept \cdot \left(\frac{1}{2} \cdot \frac{EAccept}{{KbT}^{2}} + \frac{1}{KbT}\right)\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \color{blue}{\left(EAccept \cdot \left(\frac{1}{2} \cdot \frac{EAccept}{{KbT}^{2}} + \frac{1}{KbT}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(EAccept, \color{blue}{\left(\frac{1}{2} \cdot \frac{EAccept}{{KbT}^{2}} + \frac{1}{KbT}\right)}\right)\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(EAccept, \left(\frac{1}{KbT} + \color{blue}{\frac{1}{2} \cdot \frac{EAccept}{{KbT}^{2}}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(EAccept, \mathsf{+.f64}\left(\left(\frac{1}{KbT}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{EAccept}{{KbT}^{2}}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(EAccept, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, KbT\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{EAccept}{{KbT}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(EAccept, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, KbT\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{EAccept}{{KbT}^{2}}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(EAccept, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, KbT\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(EAccept, \color{blue}{\left({KbT}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(EAccept, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, KbT\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(EAccept, \left(KbT \cdot \color{blue}{KbT}\right)\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 22.1%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(EAccept, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, KbT\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(EAccept, \mathsf{*.f64}\left(KbT, \color{blue}{KbT}\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 22.1%

       \[\leadsto \frac{NaChar}{\color{blue}{2 + EAccept \cdot \left(\frac{1}{KbT} + 0.5 \cdot \frac{EAccept}{KbT \cdot KbT}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -8000000000:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 8.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{NaChar}{2 + EAccept \cdot \left(\frac{1}{KbT} + 0.5 \cdot \frac{EAccept}{KbT \cdot KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.0% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -2.25 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(Vef + mu\right) + \left(EDonor - Ec\right)}{KbT}}\\ \mathbf{elif}\;KbT \leq 6.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{EAccept}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -2.25e+79)
     t_0
     (if (<= KbT 2.6e-195)
       (/ NdChar (+ 2.0 (/ (+ (+ Vef mu) (- EDonor Ec)) KbT)))
       (if (<= KbT 6.5e-20) (/ NaChar (+ 2.0 (/ EAccept 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 <= -2.25e+79) {
		tmp = t_0;
	} else if (KbT <= 2.6e-195) {
		tmp = NdChar / (2.0 + (((Vef + mu) + (EDonor - Ec)) / KbT));
	} else if (KbT <= 6.5e-20) {
		tmp = NaChar / (2.0 + (EAccept / 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 <= (-2.25d+79)) then
        tmp = t_0
    else if (kbt <= 2.6d-195) then
        tmp = ndchar / (2.0d0 + (((vef + mu) + (edonor - ec)) / kbt))
    else if (kbt <= 6.5d-20) then
        tmp = nachar / (2.0d0 + (eaccept / 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 <= -2.25e+79) {
		tmp = t_0;
	} else if (KbT <= 2.6e-195) {
		tmp = NdChar / (2.0 + (((Vef + mu) + (EDonor - Ec)) / KbT));
	} else if (KbT <= 6.5e-20) {
		tmp = NaChar / (2.0 + (EAccept / 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 <= -2.25e+79:
		tmp = t_0
	elif KbT <= 2.6e-195:
		tmp = NdChar / (2.0 + (((Vef + mu) + (EDonor - Ec)) / KbT))
	elif KbT <= 6.5e-20:
		tmp = NaChar / (2.0 + (EAccept / 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 <= -2.25e+79)
		tmp = t_0;
	elseif (KbT <= 2.6e-195)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Vef + mu) + Float64(EDonor - Ec)) / KbT)));
	elseif (KbT <= 6.5e-20)
		tmp = Float64(NaChar / Float64(2.0 + Float64(EAccept / 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 <= -2.25e+79)
		tmp = t_0;
	elseif (KbT <= 2.6e-195)
		tmp = NdChar / (2.0 + (((Vef + mu) + (EDonor - Ec)) / KbT));
	elseif (KbT <= 6.5e-20)
		tmp = NaChar / (2.0 + (EAccept / 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, -2.25e+79], t$95$0, If[LessEqual[KbT, 2.6e-195], N[(NdChar / N[(2.0 + N[(N[(N[(Vef + mu), $MachinePrecision] + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 6.5e-20], N[(NaChar / N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -2.24999999999999997e79 or 6.50000000000000032e-20 < 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 54.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 54.0%

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

   if -2.24999999999999997e79 < KbT < 2.6000000000000002e-195

   1. Initial program 100.0%

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

   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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   10. Simplified 27.1%

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

   11. Taylor expanded in KbT around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right), \color{blue}{KbT}\right)\right)\right) \]

       2. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(\left(Ec - EDonor\right) - \left(Vef + mu\right)\right), KbT\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(Ec - EDonor\right), \left(Vef + mu\right)\right), KbT\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(Ec, EDonor\right), \left(Vef + mu\right)\right), KbT\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(Ec, EDonor\right), \left(mu + Vef\right)\right), KbT\right)\right)\right) \]

       6. +-lowering-+.f64 23.9%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(Ec, EDonor\right), \mathsf{+.f64}\left(mu, Vef\right)\right), KbT\right)\right)\right) \]

   13. Simplified 23.9%

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

   if 2.6000000000000002e-195 < KbT < 6.50000000000000032e-20

   1. Initial program 100.0%

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

   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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 43.1%

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

   10. Simplified 43.1%

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

   11. Taylor expanded in EAccept around 0

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + \frac{EAccept}{KbT}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right) \]

       2. /-lowering-/.f64 16.6%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, \color{blue}{KbT}\right)\right)\right) \]

   13. Simplified 16.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.25 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(Vef + mu\right) + \left(EDonor - Ec\right)}{KbT}}\\ \mathbf{elif}\;KbT \leq 6.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{EAccept}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.9% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -4.1 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 4.4 \cdot 10^{-192}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(EDonor \cdot EDonor\right)}{KbT \cdot KbT}}\\ \mathbf{elif}\;KbT \leq 1.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{EAccept}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -4.1e-88)
     t_0
     (if (<= KbT 4.4e-192)
       (/ NdChar (/ (* 0.5 (* EDonor EDonor)) (* KbT KbT)))
       (if (<= KbT 1.5e-18) (/ NaChar (+ 2.0 (/ EAccept 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 <= -4.1e-88) {
		tmp = t_0;
	} else if (KbT <= 4.4e-192) {
		tmp = NdChar / ((0.5 * (EDonor * EDonor)) / (KbT * KbT));
	} else if (KbT <= 1.5e-18) {
		tmp = NaChar / (2.0 + (EAccept / 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 <= (-4.1d-88)) then
        tmp = t_0
    else if (kbt <= 4.4d-192) then
        tmp = ndchar / ((0.5d0 * (edonor * edonor)) / (kbt * kbt))
    else if (kbt <= 1.5d-18) then
        tmp = nachar / (2.0d0 + (eaccept / 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 <= -4.1e-88) {
		tmp = t_0;
	} else if (KbT <= 4.4e-192) {
		tmp = NdChar / ((0.5 * (EDonor * EDonor)) / (KbT * KbT));
	} else if (KbT <= 1.5e-18) {
		tmp = NaChar / (2.0 + (EAccept / 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 <= -4.1e-88:
		tmp = t_0
	elif KbT <= 4.4e-192:
		tmp = NdChar / ((0.5 * (EDonor * EDonor)) / (KbT * KbT))
	elif KbT <= 1.5e-18:
		tmp = NaChar / (2.0 + (EAccept / 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 <= -4.1e-88)
		tmp = t_0;
	elseif (KbT <= 4.4e-192)
		tmp = Float64(NdChar / Float64(Float64(0.5 * Float64(EDonor * EDonor)) / Float64(KbT * KbT)));
	elseif (KbT <= 1.5e-18)
		tmp = Float64(NaChar / Float64(2.0 + Float64(EAccept / 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 <= -4.1e-88)
		tmp = t_0;
	elseif (KbT <= 4.4e-192)
		tmp = NdChar / ((0.5 * (EDonor * EDonor)) / (KbT * KbT));
	elseif (KbT <= 1.5e-18)
		tmp = NaChar / (2.0 + (EAccept / 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, -4.1e-88], t$95$0, If[LessEqual[KbT, 4.4e-192], N[(NdChar / N[(N[(0.5 * N[(EDonor * EDonor), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.5e-18], N[(NaChar / N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.1000000000000001e-88 or 1.49999999999999991e-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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   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 45.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 45.4%

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

   if -4.1000000000000001e-88 < KbT < 4.40000000000000011e-192

   1. Initial program 100.0%

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

   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 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   10. Simplified 29.5%

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

   11. Taylor expanded in EDonor around inf

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\frac{1}{2} \cdot \frac{{EDonor}^{2}}{{KbT}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \left(\frac{\frac{1}{2} \cdot {EDonor}^{2}}{\color{blue}{{KbT}^{2}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {EDonor}^{2}\right), \color{blue}{\left({KbT}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({EDonor}^{2}\right)\right), \left({\color{blue}{KbT}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(EDonor \cdot EDonor\right)\right), \left({KbT}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(EDonor, EDonor\right)\right), \left({KbT}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(EDonor, EDonor\right)\right), \left(KbT \cdot \color{blue}{KbT}\right)\right)\right) \]

       7. *-lowering-*.f64 20.1%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(EDonor, EDonor\right)\right), \mathsf{*.f64}\left(KbT, \color{blue}{KbT}\right)\right)\right) \]

   13. Simplified 20.1%

       \[\leadsto \frac{NdChar}{\color{blue}{\frac{0.5 \cdot \left(EDonor \cdot EDonor\right)}{KbT \cdot KbT}}} \]

   if 4.40000000000000011e-192 < KbT < 1.49999999999999991e-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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 43.1%

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

   10. Simplified 43.1%

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

   11. Taylor expanded in EAccept around 0

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + \frac{EAccept}{KbT}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right) \]

       2. /-lowering-/.f64 16.6%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EAccept, \color{blue}{KbT}\right)\right)\right) \]

   13. Simplified 16.6%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.1 \cdot 10^{-88}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 4.4 \cdot 10^{-192}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(EDonor \cdot EDonor\right)}{KbT \cdot KbT}}\\ \mathbf{elif}\;KbT \leq 1.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{EAccept}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 31.6% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -8.2 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 7.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Vef \cdot Vef}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -8.2e-17)
     t_0
     (if (<= KbT 7.5e-124)
       (/ (* 2.0 (* NdChar (* KbT KbT))) (* Vef Vef))
       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 <= -8.2e-17) {
		tmp = t_0;
	} else if (KbT <= 7.5e-124) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef);
	} 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 <= (-8.2d-17)) then
        tmp = t_0
    else if (kbt <= 7.5d-124) then
        tmp = (2.0d0 * (ndchar * (kbt * kbt))) / (vef * vef)
    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 <= -8.2e-17) {
		tmp = t_0;
	} else if (KbT <= 7.5e-124) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef);
	} 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 <= -8.2e-17:
		tmp = t_0
	elif KbT <= 7.5e-124:
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef)
	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 <= -8.2e-17)
		tmp = t_0;
	elseif (KbT <= 7.5e-124)
		tmp = Float64(Float64(2.0 * Float64(NdChar * Float64(KbT * KbT))) / Float64(Vef * Vef));
	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 <= -8.2e-17)
		tmp = t_0;
	elseif (KbT <= 7.5e-124)
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef);
	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, -8.2e-17], t$95$0, If[LessEqual[KbT, 7.5e-124], N[(N[(2.0 * N[(NdChar * N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Vef * Vef), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -8.2000000000000001e-17 or 7.4999999999999996e-124 < 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 44.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 44.9%

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

   if -8.2000000000000001e-17 < KbT < 7.4999999999999996e-124

   1. Initial program 100.0%

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

   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 65.6%

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

   7. Simplified 65.6%

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

   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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   10. Simplified 27.3%

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

   11. Taylor expanded in Vef around inf

       \[\leadsto \color{blue}{2 \cdot \frac{{KbT}^{2} \cdot NdChar}{{Vef}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}{\color{blue}{{Vef}^{2}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({KbT}^{2} \cdot NdChar\right)\right), \color{blue}{\left({Vef}^{2}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({KbT}^{2} \cdot NdChar\right)\right), \left({\color{blue}{Vef}}^{2}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(NdChar \cdot {KbT}^{2}\right)\right), \left({Vef}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(NdChar, \left({KbT}^{2}\right)\right)\right), \left({Vef}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(NdChar, \left(KbT \cdot KbT\right)\right)\right), \left({Vef}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(NdChar, \mathsf{*.f64}\left(KbT, KbT\right)\right)\right), \left({Vef}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(NdChar, \mathsf{*.f64}\left(KbT, KbT\right)\right)\right), \left(Vef \cdot \color{blue}{Vef}\right)\right) \]

       9. *-lowering-*.f64 22.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(NdChar, \mathsf{*.f64}\left(KbT, KbT\right)\right)\right), \mathsf{*.f64}\left(Vef, \color{blue}{Vef}\right)\right) \]

   13. Simplified 22.0%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Vef \cdot Vef}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -8.2 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 7.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Vef \cdot Vef}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.1% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.3 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(Ec \cdot Ec\right)}{KbT \cdot KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.3e-89)
     t_0
     (if (<= KbT 5e-128) (/ NdChar (/ (* 0.5 (* Ec Ec)) (* 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.3e-89) {
		tmp = t_0;
	} else if (KbT <= 5e-128) {
		tmp = NdChar / ((0.5 * (Ec * Ec)) / (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) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.3d-89)) then
        tmp = t_0
    else if (kbt <= 5d-128) then
        tmp = ndchar / ((0.5d0 * (ec * ec)) / (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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.3e-89) {
		tmp = t_0;
	} else if (KbT <= 5e-128) {
		tmp = NdChar / ((0.5 * (Ec * Ec)) / (KbT * 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 <= -1.3e-89:
		tmp = t_0
	elif KbT <= 5e-128:
		tmp = NdChar / ((0.5 * (Ec * Ec)) / (KbT * 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 <= -1.3e-89)
		tmp = t_0;
	elseif (KbT <= 5e-128)
		tmp = Float64(NdChar / Float64(Float64(0.5 * Float64(Ec * Ec)) / Float64(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 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.3e-89)
		tmp = t_0;
	elseif (KbT <= 5e-128)
		tmp = NdChar / ((0.5 * (Ec * Ec)) / (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[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.3e-89], t$95$0, If[LessEqual[KbT, 5e-128], N[(NdChar / N[(N[(0.5 * N[(Ec * Ec), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -1.2999999999999999e-89 or 5.0000000000000001e-128 < 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 41.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 41.1%

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

   if -1.2999999999999999e-89 < KbT < 5.0000000000000001e-128

   1. Initial program 100.0%

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

   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 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   10. Simplified 28.3%

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

   11. Taylor expanded in Ec around inf

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\frac{1}{2} \cdot \frac{{Ec}^{2}}{{KbT}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \left(\frac{\frac{1}{2} \cdot {Ec}^{2}}{\color{blue}{{KbT}^{2}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {Ec}^{2}\right), \color{blue}{\left({KbT}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({Ec}^{2}\right)\right), \left({\color{blue}{KbT}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(Ec \cdot Ec\right)\right), \left({KbT}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(Ec, Ec\right)\right), \left({KbT}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(Ec, Ec\right)\right), \left(KbT \cdot \color{blue}{KbT}\right)\right)\right) \]

       7. *-lowering-*.f64 23.1%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(Ec, Ec\right)\right), \mathsf{*.f64}\left(KbT, \color{blue}{KbT}\right)\right)\right) \]

   13. Simplified 23.1%

       \[\leadsto \frac{NdChar}{\color{blue}{\frac{0.5 \cdot \left(Ec \cdot Ec\right)}{KbT \cdot KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.3 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(Ec \cdot Ec\right)}{KbT \cdot KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 22.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.12 \cdot 10^{-145}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 4.1 \cdot 10^{+113}:\\ \;\;\;\;\frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -1.12e-145)
   (* NdChar 0.5)
   (if (<= NdChar 4.1e+113) (/ NaChar 2.0) (* NdChar 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 (NdChar <= -1.12e-145) {
		tmp = NdChar * 0.5;
	} else if (NdChar <= 4.1e+113) {
		tmp = NaChar / 2.0;
	} else {
		tmp = NdChar * 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 (ndchar <= (-1.12d-145)) then
        tmp = ndchar * 0.5d0
    else if (ndchar <= 4.1d+113) then
        tmp = nachar / 2.0d0
    else
        tmp = ndchar * 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 (NdChar <= -1.12e-145) {
		tmp = NdChar * 0.5;
	} else if (NdChar <= 4.1e+113) {
		tmp = NaChar / 2.0;
	} else {
		tmp = NdChar * 0.5;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -1.12e-145:
		tmp = NdChar * 0.5
	elif NdChar <= 4.1e+113:
		tmp = NaChar / 2.0
	else:
		tmp = NdChar * 0.5
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -1.12e-145)
		tmp = Float64(NdChar * 0.5);
	elseif (NdChar <= 4.1e+113)
		tmp = Float64(NaChar / 2.0);
	else
		tmp = Float64(NdChar * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -1.12e-145)
		tmp = NdChar * 0.5;
	elseif (NdChar <= 4.1e+113)
		tmp = NaChar / 2.0;
	else
		tmp = NdChar * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -1.12e-145], N[(NdChar * 0.5), $MachinePrecision], If[LessEqual[NdChar, 4.1e+113], N[(NaChar / 2.0), $MachinePrecision], N[(NdChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.12000000000000001e-145 or 4.09999999999999993e113 < 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 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 29.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 29.6%

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

   8. Taylor expanded in NaChar around 0

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

      1. *-commutative N/A

         \[\leadsto NdChar \cdot \color{blue}{\frac{1}{2}} \]

      2. *-lowering-*.f64 25.8%

         \[\leadsto \mathsf{*.f64}\left(NdChar, \color{blue}{\frac{1}{2}}\right) \]

   10. Simplified 25.8%

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

   if -1.12000000000000001e-145 < NdChar < 4.09999999999999993e113

   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 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. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f64 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{2}\right) \]
   9. Step-by-step derivation

   10. Simplified 30.6%

       \[\leadsto \frac{NaChar}{\color{blue}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 27.8% 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 31.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

7. Simplified 31.3%

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

8. Final simplification 31.3%

   \[\leadsto 0.5 \cdot \left(NdChar + NaChar\right) \]
9. Add Preprocessing

Alternative 21: 18.1% accurate, 76.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}}} \]
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 31.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

7. Simplified 31.3%

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

8. Taylor expanded in NaChar around 0

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

   1. *-commutative N/A

      \[\leadsto NdChar \cdot \color{blue}{\frac{1}{2}} \]

   2. *-lowering-*.f64 19.4%

      \[\leadsto \mathsf{*.f64}\left(NdChar, \color{blue}{\frac{1}{2}}\right) \]

10. Simplified 19.4%

    \[\leadsto \color{blue}{NdChar \cdot 0.5} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024149 
(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))))))