Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 21.6s

Alternatives: 19

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = -6.758934001644053e+41
  2. Ec = -3.933238538563548e-211
  3. Vef = 4.5957171737107014e+163
  4. EDonor = -1.7160130094166495e+183
  5. mu = 1.8041619464457133e-279
  6. KbT = 8.380780543598958e+73
  7. NaChar = -6.817083025198011e+41
  8. Ev = -1.6504762322024825e+295
  9. EAccept = 7.652042915879465e-128

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

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. +-lowering-+.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}} + 1}\\ t_1 := t\_0 + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -4.2 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq 10.5:\\ \;\;\;\;t\_0 + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (+ Vef (- (+ EDonor mu) Ec)) KbT)) 1.0)))
        (t_1 (+ t_0 (/ NaChar (+ (exp (/ Vef KbT)) 1.0)))))
   (if (<= Vef -4.2e+89)
     t_1
     (if (<= Vef 10.5) (+ t_0 (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))) t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp(((Vef + ((EDonor + mu) - Ec)) / KbT)) + 1.0);
	double t_1 = t_0 + (NaChar / (exp((Vef / KbT)) + 1.0));
	double tmp;
	if (Vef <= -4.2e+89) {
		tmp = t_1;
	} else if (Vef <= 10.5) {
		tmp = t_0 + (NaChar / (exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (exp(((vef + ((edonor + mu) - ec)) / kbt)) + 1.0d0)
    t_1 = t_0 + (nachar / (exp((vef / kbt)) + 1.0d0))
    if (vef <= (-4.2d+89)) then
        tmp = t_1
    else if (vef <= 10.5d0) then
        tmp = t_0 + (nachar / (exp((eaccept / kbt)) + 1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp(((Vef + ((EDonor + mu) - Ec)) / KbT)) + 1.0);
	double t_1 = t_0 + (NaChar / (Math.exp((Vef / KbT)) + 1.0));
	double tmp;
	if (Vef <= -4.2e+89) {
		tmp = t_1;
	} else if (Vef <= 10.5) {
		tmp = t_0 + (NaChar / (Math.exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((Vef + ((EDonor + mu) - Ec)) / KbT)) + 1.0)
	t_1 = t_0 + (NaChar / (math.exp((Vef / KbT)) + 1.0))
	tmp = 0
	if Vef <= -4.2e+89:
		tmp = t_1
	elif Vef <= 10.5:
		tmp = t_0 + (NaChar / (math.exp((EAccept / KbT)) + 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(Vef + Float64(Float64(EDonor + mu) - Ec)) / KbT)) + 1.0))
	t_1 = Float64(t_0 + Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)))
	tmp = 0.0
	if (Vef <= -4.2e+89)
		tmp = t_1;
	elseif (Vef <= 10.5)
		tmp = Float64(t_0 + Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp(((Vef + ((EDonor + mu) - Ec)) / KbT)) + 1.0);
	t_1 = t_0 + (NaChar / (exp((Vef / KbT)) + 1.0));
	tmp = 0.0;
	if (Vef <= -4.2e+89)
		tmp = t_1;
	elseif (Vef <= 10.5)
		tmp = t_0 + (NaChar / (exp((EAccept / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(Vef + N[(N[(EDonor + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -4.2e+89], t$95$1, If[LessEqual[Vef, 10.5], N[(t$95$0 + N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -4.19999999999999972e89 or 10.5 < Vef

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 84.6%

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

   7. Simplified 84.6%

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

   if -4.19999999999999972e89 < Vef < 10.5

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 74.4%

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

   7. Simplified 74.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -4.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 10.5:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -2.8 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT)) 1.0))))
   (if (<= NaChar -2.8e+95)
     t_0
     (if (<= NaChar 2.9e-58)
       (+
        (/ NdChar (+ (exp (/ (+ Vef (- (+ EDonor mu) Ec)) KbT)) 1.0))
        (/ NaChar (+ (exp (/ EAccept 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 + (Ev + EAccept)) - mu) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -2.8e+95) {
		tmp = t_0;
	} else if (NaChar <= 2.9e-58) {
		tmp = (NdChar / (exp(((Vef + ((EDonor + mu) - Ec)) / KbT)) + 1.0)) + (NaChar / (exp((EAccept / 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 + (ev + eaccept)) - mu) / kbt)) + 1.0d0)
    if (nachar <= (-2.8d+95)) then
        tmp = t_0
    else if (nachar <= 2.9d-58) then
        tmp = (ndchar / (exp(((vef + ((edonor + mu) - ec)) / kbt)) + 1.0d0)) + (nachar / (exp((eaccept / 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 + (Ev + EAccept)) - mu) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -2.8e+95) {
		tmp = t_0;
	} else if (NaChar <= 2.9e-58) {
		tmp = (NdChar / (Math.exp(((Vef + ((EDonor + mu) - Ec)) / KbT)) + 1.0)) + (NaChar / (Math.exp((EAccept / 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 + (Ev + EAccept)) - mu) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -2.8e+95:
		tmp = t_0
	elif NaChar <= 2.9e-58:
		tmp = (NdChar / (math.exp(((Vef + ((EDonor + mu) - Ec)) / KbT)) + 1.0)) + (NaChar / (math.exp((EAccept / 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(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -2.8e+95)
		tmp = t_0;
	elseif (NaChar <= 2.9e-58)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Vef + Float64(Float64(EDonor + mu) - Ec)) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(EAccept / 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 + (Ev + EAccept)) - mu) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -2.8e+95)
		tmp = t_0;
	elseif (NaChar <= 2.9e-58)
		tmp = (NdChar / (exp(((Vef + ((EDonor + mu) - Ec)) / KbT)) + 1.0)) + (NaChar / (exp((EAccept / 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[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -2.8e+95], t$95$0, If[LessEqual[NaChar, 2.9e-58], N[(N[(NdChar / N[(N[Exp[N[(N[(Vef + N[(N[(EDonor + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -2.7999999999999998e95 or 2.8999999999999999e-58 < NaChar

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-lowering-+.f64 78.3%

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

   7. Simplified 78.3%

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

   if -2.7999999999999998e95 < NaChar < 2.8999999999999999e-58

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 77.4%

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

   7. Simplified 77.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.8 \cdot 10^{+95}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NaChar / (exp(((Vef - ((mu - EAccept) - Ev)) / KbT)) + 1.0)) + (NdChar / (exp(((Vef + ((EDonor + mu) - 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[(N[(mu - EAccept), $MachinePrecision] - Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(Vef + N[(N[(EDonor + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

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

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 5: 66.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{if}\;NdChar \leq -1.9 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 6.3 \cdot 10^{+181}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (- (+ mu (+ Vef EDonor)) Ec) KbT)) 1.0))))
   (if (<= NdChar -1.9e-157)
     t_0
     (if (<= NdChar 6.3e+181)
       (/ NaChar (+ (exp (/ (- (+ Vef (+ Ev EAccept)) 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((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	double tmp;
	if (NdChar <= -1.9e-157) {
		tmp = t_0;
	} else if (NdChar <= 6.3e+181) {
		tmp = NaChar / (exp((((Vef + (Ev + EAccept)) - 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((((mu + (vef + edonor)) - ec) / kbt)) + 1.0d0)
    if (ndchar <= (-1.9d-157)) then
        tmp = t_0
    else if (ndchar <= 6.3d+181) then
        tmp = nachar / (exp((((vef + (ev + eaccept)) - 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((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	double tmp;
	if (NdChar <= -1.9e-157) {
		tmp = t_0;
	} else if (NdChar <= 6.3e+181) {
		tmp = NaChar / (Math.exp((((Vef + (Ev + EAccept)) - 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((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0)
	tmp = 0
	if NdChar <= -1.9e-157:
		tmp = t_0
	elif NdChar <= 6.3e+181:
		tmp = NaChar / (math.exp((((Vef + (Ev + EAccept)) - 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(NdChar / Float64(exp(Float64(Float64(Float64(mu + Float64(Vef + EDonor)) - Ec) / KbT)) + 1.0))
	tmp = 0.0
	if (NdChar <= -1.9e-157)
		tmp = t_0;
	elseif (NdChar <= 6.3e+181)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - 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((((mu + (Vef + EDonor)) - Ec) / KbT)) + 1.0);
	tmp = 0.0;
	if (NdChar <= -1.9e-157)
		tmp = t_0;
	elseif (NdChar <= 6.3e+181)
		tmp = NaChar / (exp((((Vef + (Ev + EAccept)) - 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[(NdChar / N[(N[Exp[N[(N[(N[(mu + N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.9e-157], t$95$0, If[LessEqual[NdChar, 6.3e+181], N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.9000000000000001e-157 or 6.3000000000000003e181 < NdChar

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 73.7%

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

   7. Simplified 73.7%

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

   if -1.9000000000000001e-157 < NdChar < 6.3000000000000003e181

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-lowering-+.f64 77.1%

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

   7. Simplified 77.1%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.9 \cdot 10^{-157}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 6.3 \cdot 10^{+181}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + \left(Vef + EDonor\right)\right) - Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4.8 \cdot 10^{+244}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 3.6 \cdot 10^{+218}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - -0.25 \cdot \left(Ec \cdot \frac{NdChar}{KbT}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4.8e+244)
   (+ (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (/ NaChar 2.0))
   (if (<= KbT 3.6e+218)
     (/ NaChar (+ (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT)) 1.0))
     (- (* 0.5 (+ NdChar NaChar)) (* -0.25 (* Ec (/ NdChar KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.8e+244) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	} else if (KbT <= 3.6e+218) {
		tmp = NaChar / (exp((((Vef + (Ev + EAccept)) - mu) / KbT)) + 1.0);
	} else {
		tmp = (0.5 * (NdChar + NaChar)) - (-0.25 * (Ec * (NdChar / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-4.8d+244)) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar / 2.0d0)
    else if (kbt <= 3.6d+218) then
        tmp = nachar / (exp((((vef + (ev + eaccept)) - mu) / kbt)) + 1.0d0)
    else
        tmp = (0.5d0 * (ndchar + nachar)) - ((-0.25d0) * (ec * (ndchar / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.8e+244) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	} else if (KbT <= 3.6e+218) {
		tmp = NaChar / (Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)) + 1.0);
	} else {
		tmp = (0.5 * (NdChar + NaChar)) - (-0.25 * (Ec * (NdChar / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -4.8e+244:
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0)
	elif KbT <= 3.6e+218:
		tmp = NaChar / (math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)) + 1.0)
	else:
		tmp = (0.5 * (NdChar + NaChar)) - (-0.25 * (Ec * (NdChar / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4.8e+244)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar / 2.0));
	elseif (KbT <= 3.6e+218)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)) + 1.0));
	else
		tmp = Float64(Float64(0.5 * Float64(NdChar + NaChar)) - Float64(-0.25 * Float64(Ec * Float64(NdChar / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -4.8e+244)
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / 2.0);
	elseif (KbT <= 3.6e+218)
		tmp = NaChar / (exp((((Vef + (Ev + EAccept)) - mu) / KbT)) + 1.0);
	else
		tmp = (0.5 * (NdChar + NaChar)) - (-0.25 * (Ec * (NdChar / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4.8e+244], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 3.6e+218], N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision] - N[(-0.25 * N[(Ec * N[(NdChar / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.79999999999999975e244

   1. Initial program 99.9%

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

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

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf

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

   7. Simplified 86.2%

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

   8. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 82.3%

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

   10. Simplified 82.3%

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

   if -4.79999999999999975e244 < KbT < 3.59999999999999991e218

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-lowering-+.f64 67.3%

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

   7. Simplified 67.3%

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

   if 3.59999999999999991e218 < KbT

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \color{blue}{\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right)\right) + \frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \left(\color{blue}{\frac{1}{2}} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)\right)\right) + \left(\frac{1}{2} \cdot \color{blue}{NaChar} + \frac{1}{2} \cdot NdChar\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)\right)\right) + \left(\color{blue}{\frac{1}{2} \cdot NaChar} + \frac{1}{2} \cdot NdChar\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)}\right) \]

   7. Simplified 58.7%

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

   8. Taylor expanded in Ec around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left(-1 \cdot \frac{Ec \cdot NdChar}{KbT}\right)}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(\mathsf{neg}\left(\frac{Ec \cdot NdChar}{KbT}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(0 - \frac{Ec \cdot NdChar}{KbT}\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{\_.f64}\left(0, \left(\frac{Ec \cdot NdChar}{KbT}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{\_.f64}\left(0, \left(Ec \cdot \frac{NdChar}{KbT}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(Ec, \left(\frac{NdChar}{KbT}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

      6. /-lowering-/.f64 79.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(Ec, \mathsf{/.f64}\left(NdChar, KbT\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

   10. Simplified 79.7%

       \[\leadsto -0.25 \cdot \color{blue}{\left(0 - Ec \cdot \frac{NdChar}{KbT}\right)} + 0.5 \cdot \left(NaChar + NdChar\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.8 \cdot 10^{+244}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 3.6 \cdot 10^{+218}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - -0.25 \cdot \left(Ec \cdot \frac{NdChar}{KbT}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;NdChar \leq -3.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{elif}\;NdChar \leq 2.85 \cdot 10^{+189}:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0)))
   (if (<= NdChar -3.2e-156)
     (/ NdChar t_0)
     (if (<= NdChar 2.85e+189)
       (/ NaChar t_0)
       (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((Vef / KbT)) + 1.0;
	double tmp;
	if (NdChar <= -3.2e-156) {
		tmp = NdChar / t_0;
	} else if (NdChar <= 2.85e+189) {
		tmp = NaChar / t_0;
	} else {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    if (ndchar <= (-3.2d-156)) then
        tmp = ndchar / t_0
    else if (ndchar <= 2.85d+189) then
        tmp = nachar / t_0
    else
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp((Vef / KbT)) + 1.0;
	double tmp;
	if (NdChar <= -3.2e-156) {
		tmp = NdChar / t_0;
	} else if (NdChar <= 2.85e+189) {
		tmp = NaChar / t_0;
	} else {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	tmp = 0
	if NdChar <= -3.2e-156:
		tmp = NdChar / t_0
	elif NdChar <= 2.85e+189:
		tmp = NaChar / t_0
	else:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	tmp = 0.0
	if (NdChar <= -3.2e-156)
		tmp = Float64(NdChar / t_0);
	elseif (NdChar <= 2.85e+189)
		tmp = Float64(NaChar / t_0);
	else
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp((Vef / KbT)) + 1.0;
	tmp = 0.0;
	if (NdChar <= -3.2e-156)
		tmp = NdChar / t_0;
	elseif (NdChar <= 2.85e+189)
		tmp = NaChar / t_0;
	else
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[NdChar, -3.2e-156], N[(NdChar / t$95$0), $MachinePrecision], If[LessEqual[NdChar, 2.85e+189], N[(NaChar / t$95$0), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -3.19999999999999982e-156

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 72.0%

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

   7. Simplified 72.0%

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

   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 51.5%

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

   10. Simplified 51.5%

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

   if -3.19999999999999982e-156 < NdChar < 2.8500000000000001e189

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 49.7%

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

   10. Simplified 49.7%

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

   if 2.8500000000000001e189 < NdChar

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 65.5%

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

   10. Simplified 65.5%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -3.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 2.85 \cdot 10^{+189}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{if}\;NdChar \leq -3.8 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 2.85 \cdot 10^{+189}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))))
   (if (<= NdChar -3.8e-156)
     t_0
     (if (<= NdChar 2.85e+189) (/ NaChar (+ (exp (/ Vef 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 (NdChar <= -3.8e-156) {
		tmp = t_0;
	} else if (NdChar <= 2.85e+189) {
		tmp = NaChar / (exp((Vef / 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 (ndchar <= (-3.8d-156)) then
        tmp = t_0
    else if (ndchar <= 2.85d+189) then
        tmp = nachar / (exp((vef / 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 (NdChar <= -3.8e-156) {
		tmp = t_0;
	} else if (NdChar <= 2.85e+189) {
		tmp = NaChar / (Math.exp((Vef / 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 NdChar <= -3.8e-156:
		tmp = t_0
	elif NdChar <= 2.85e+189:
		tmp = NaChar / (math.exp((Vef / 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 (NdChar <= -3.8e-156)
		tmp = t_0;
	elseif (NdChar <= 2.85e+189)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / 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 (NdChar <= -3.8e-156)
		tmp = t_0;
	elseif (NdChar <= 2.85e+189)
		tmp = NaChar / (exp((Vef / 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[NdChar, -3.8e-156], t$95$0, If[LessEqual[NdChar, 2.85e+189], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -3.80000000000000008e-156 or 2.8500000000000001e189 < NdChar

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 48.1%

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

   10. Simplified 48.1%

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

   if -3.80000000000000008e-156 < NdChar < 2.8500000000000001e189

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 49.7%

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

   10. Simplified 49.7%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -3.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 2.85 \cdot 10^{+189}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -7.2 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.02 \cdot 10^{+148}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(Vef \cdot -0.25\right) \cdot \left(\frac{NdChar}{KbT} + \frac{NaChar}{KbT}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -7.2e+92)
     t_0
     (if (<= KbT 1.02e+148)
       (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
       (+ t_0 (* (* Vef -0.25) (+ (/ NdChar KbT) (/ NaChar KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -7.2e+92) {
		tmp = t_0;
	} else if (KbT <= 1.02e+148) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = t_0 + ((Vef * -0.25) * ((NdChar / KbT) + (NaChar / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-7.2d+92)) then
        tmp = t_0
    else if (kbt <= 1.02d+148) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = t_0 + ((vef * (-0.25d0)) * ((ndchar / kbt) + (nachar / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -7.2e+92) {
		tmp = t_0;
	} else if (KbT <= 1.02e+148) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = t_0 + ((Vef * -0.25) * ((NdChar / KbT) + (NaChar / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -7.2e+92:
		tmp = t_0
	elif KbT <= 1.02e+148:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = t_0 + ((Vef * -0.25) * ((NdChar / KbT) + (NaChar / KbT)))
	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 <= -7.2e+92)
		tmp = t_0;
	elseif (KbT <= 1.02e+148)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(t_0 + Float64(Float64(Vef * -0.25) * Float64(Float64(NdChar / KbT) + Float64(NaChar / KbT))));
	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 <= -7.2e+92)
		tmp = t_0;
	elseif (KbT <= 1.02e+148)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = t_0 + ((Vef * -0.25) * ((NdChar / KbT) + (NaChar / KbT)));
	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, -7.2e+92], t$95$0, If[LessEqual[KbT, 1.02e+148], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(Vef * -0.25), $MachinePrecision] * N[(N[(NdChar / KbT), $MachinePrecision] + N[(NaChar / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -7.2e92

   1. Initial program 99.9%

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

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

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 56.4%

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

   7. Simplified 56.4%

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

   if -7.2e92 < KbT < 1.02e148

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 43.5%

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

   10. Simplified 43.5%

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

   if 1.02e148 < KbT

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \color{blue}{\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right)\right) + \frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \left(\color{blue}{\frac{1}{2}} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)\right)\right) + \left(\frac{1}{2} \cdot \color{blue}{NaChar} + \frac{1}{2} \cdot NdChar\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)\right)\right) + \left(\color{blue}{\frac{1}{2} \cdot NaChar} + \frac{1}{2} \cdot NdChar\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)}\right) \]

   7. Simplified 42.6%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left(Vef \cdot \left(\frac{NaChar}{KbT} + \frac{NdChar}{KbT}\right)\right)\right)}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{-1}{4} \cdot Vef\right) \cdot \left(\frac{NaChar}{KbT} + \frac{NdChar}{KbT}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{4} \cdot Vef\right), \left(\frac{NaChar}{KbT} + \frac{NdChar}{KbT}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, Vef\right), \left(\frac{NaChar}{KbT} + \frac{NdChar}{KbT}\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, Vef\right), \mathsf{+.f64}\left(\left(\frac{NaChar}{KbT}\right), \left(\frac{NdChar}{KbT}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(NaChar, KbT\right), \left(\frac{NdChar}{KbT}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

      6. /-lowering-/.f64 62.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(NaChar, KbT\right), \mathsf{/.f64}\left(NdChar, KbT\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right)\right) \]

   10. Simplified 62.0%

       \[\leadsto \color{blue}{\left(-0.25 \cdot Vef\right) \cdot \left(\frac{NaChar}{KbT} + \frac{NdChar}{KbT}\right)} + 0.5 \cdot \left(NaChar + NdChar\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -7.2 \cdot 10^{+92}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.02 \cdot 10^{+148}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) + \left(Vef \cdot -0.25\right) \cdot \left(\frac{NdChar}{KbT} + \frac{NaChar}{KbT}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Vef + EDonor\right) + \left(mu - Ec\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.72 \cdot 10^{+75}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;KbT \leq 4.3 \cdot 10^{+55}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{t\_0 + \frac{\frac{0.16666666666666666 \cdot \left(t\_0 \cdot t\_1\right)}{KbT} + 0.5 \cdot t\_1}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Vef EDonor) (- mu Ec)))
        (t_1 (* t_0 t_0))
        (t_2 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.72e+75)
     t_2
     (if (<= KbT 4.3e+55)
       (/
        NdChar
        (+
         2.0
         (/
          (+
           t_0
           (/ (+ (/ (* 0.16666666666666666 (* t_0 t_1)) KbT) (* 0.5 t_1)) KbT))
          KbT)))
       t_2))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + EDonor) + (mu - Ec);
	double t_1 = t_0 * t_0;
	double t_2 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.72e+75) {
		tmp = t_2;
	} else if (KbT <= 4.3e+55) {
		tmp = NdChar / (2.0 + ((t_0 + ((((0.16666666666666666 * (t_0 * t_1)) / KbT) + (0.5 * t_1)) / KbT)) / KbT));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (vef + edonor) + (mu - ec)
    t_1 = t_0 * t_0
    t_2 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.72d+75)) then
        tmp = t_2
    else if (kbt <= 4.3d+55) then
        tmp = ndchar / (2.0d0 + ((t_0 + ((((0.16666666666666666d0 * (t_0 * t_1)) / kbt) + (0.5d0 * t_1)) / kbt)) / kbt))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + EDonor) + (mu - Ec);
	double t_1 = t_0 * t_0;
	double t_2 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.72e+75) {
		tmp = t_2;
	} else if (KbT <= 4.3e+55) {
		tmp = NdChar / (2.0 + ((t_0 + ((((0.16666666666666666 * (t_0 * t_1)) / KbT) + (0.5 * t_1)) / KbT)) / KbT));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (Vef + EDonor) + (mu - Ec)
	t_1 = t_0 * t_0
	t_2 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.72e+75:
		tmp = t_2
	elif KbT <= 4.3e+55:
		tmp = NdChar / (2.0 + ((t_0 + ((((0.16666666666666666 * (t_0 * t_1)) / KbT) + (0.5 * t_1)) / KbT)) / KbT))
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Vef + EDonor) + Float64(mu - Ec))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.72e+75)
		tmp = t_2;
	elseif (KbT <= 4.3e+55)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(t_0 + Float64(Float64(Float64(Float64(0.16666666666666666 * Float64(t_0 * t_1)) / KbT) + Float64(0.5 * t_1)) / KbT)) / KbT)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (Vef + EDonor) + (mu - Ec);
	t_1 = t_0 * t_0;
	t_2 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.72e+75)
		tmp = t_2;
	elseif (KbT <= 4.3e+55)
		tmp = NdChar / (2.0 + ((t_0 + ((((0.16666666666666666 * (t_0 * t_1)) / KbT) + (0.5 * t_1)) / KbT)) / KbT));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.72e+75], t$95$2, If[LessEqual[KbT, 4.3e+55], N[(NdChar / N[(2.0 + N[(N[(t$95$0 + N[(N[(N[(N[(0.16666666666666666 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -1.72e75 or 4.2999999999999999e55 < KbT

   1. Initial program 99.9%

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

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

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 52.9%

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

   7. Simplified 52.9%

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

   if -1.72e75 < KbT < 4.2999999999999999e55

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \left(2 + \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \left(2 - \color{blue}{\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{3}}{KbT} + \frac{1}{2} \cdot {\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}\right), \color{blue}{KbT}\right)\right)\right) \]

   10. Simplified 34.6%

       \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\left(0 - \left(\left(EDonor + Vef\right) + \left(mu - Ec\right)\right)\right) - \frac{\frac{0.16666666666666666 \cdot \left(\left(\left(EDonor + Vef\right) + \left(mu - Ec\right)\right) \cdot \left(\left(\left(EDonor + Vef\right) + \left(mu - Ec\right)\right) \cdot \left(\left(EDonor + Vef\right) + \left(mu - Ec\right)\right)\right)\right)}{KbT} + 0.5 \cdot \left(\left(\left(EDonor + Vef\right) + \left(mu - Ec\right)\right) \cdot \left(\left(EDonor + Vef\right) + \left(mu - Ec\right)\right)\right)}{KbT}}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.72 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 4.3 \cdot 10^{+55}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) + \frac{\frac{0.16666666666666666 \cdot \left(\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right)\right)\right)}{KbT} + 0.5 \cdot \left(\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Vef + EDonor\right) + \left(mu - Ec\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -4.7 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{t\_0 + \frac{-0.5 \cdot \left(t\_0 \cdot \left(\left(Ec - mu\right) - \left(Vef + EDonor\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Vef EDonor) (- mu Ec))) (t_1 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -4.7e+39)
     t_1
     (if (<= KbT 5.2e+57)
       (/
        NdChar
        (+
         2.0
         (/
          (+ t_0 (/ (* -0.5 (* t_0 (- (- Ec mu) (+ Vef EDonor)))) KbT))
          KbT)))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + EDonor) + (mu - Ec);
	double t_1 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -4.7e+39) {
		tmp = t_1;
	} else if (KbT <= 5.2e+57) {
		tmp = NdChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((Ec - mu) - (Vef + EDonor)))) / KbT)) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (vef + edonor) + (mu - ec)
    t_1 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-4.7d+39)) then
        tmp = t_1
    else if (kbt <= 5.2d+57) then
        tmp = ndchar / (2.0d0 + ((t_0 + (((-0.5d0) * (t_0 * ((ec - mu) - (vef + edonor)))) / kbt)) / kbt))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + EDonor) + (mu - Ec);
	double t_1 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -4.7e+39) {
		tmp = t_1;
	} else if (KbT <= 5.2e+57) {
		tmp = NdChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((Ec - mu) - (Vef + EDonor)))) / KbT)) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (Vef + EDonor) + (mu - Ec)
	t_1 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -4.7e+39:
		tmp = t_1
	elif KbT <= 5.2e+57:
		tmp = NdChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((Ec - mu) - (Vef + EDonor)))) / KbT)) / KbT))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Vef + EDonor) + Float64(mu - Ec))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -4.7e+39)
		tmp = t_1;
	elseif (KbT <= 5.2e+57)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(t_0 + Float64(Float64(-0.5 * Float64(t_0 * Float64(Float64(Ec - mu) - Float64(Vef + EDonor)))) / KbT)) / KbT)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (Vef + EDonor) + (mu - Ec);
	t_1 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -4.7e+39)
		tmp = t_1;
	elseif (KbT <= 5.2e+57)
		tmp = NdChar / (2.0 + ((t_0 + ((-0.5 * (t_0 * ((Ec - mu) - (Vef + EDonor)))) / KbT)) / KbT));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -4.7e+39], t$95$1, If[LessEqual[KbT, 5.2e+57], N[(NdChar / N[(2.0 + N[(N[(t$95$0 + N[(N[(-0.5 * N[(t$95$0 * N[(N[(Ec - mu), $MachinePrecision] - N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -4.6999999999999999e39 or 5.2e57 < KbT

   1. Initial program 99.9%

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

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

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 51.2%

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

   7. Simplified 51.2%

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

   if -4.6999999999999999e39 < KbT < 5.2e57

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 68.3%

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

   7. Simplified 68.3%

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

   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 33.6%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.7 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) + \frac{-0.5 \cdot \left(\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(Ec - mu\right) - \left(Vef + EDonor\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.35 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.7 \cdot 10^{-157}:\\ \;\;\;\;\frac{NdChar}{2 + mu \cdot \left(\frac{1}{KbT} + \frac{mu \cdot 0.5}{KbT \cdot KbT}\right)}\\ \mathbf{elif}\;KbT \leq 4.9 \cdot 10^{-45}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\frac{EDonor}{KbT} + 2\right) + Vef \cdot \left(\frac{1}{KbT} + \frac{mu}{Vef \cdot KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.35e+39)
     t_0
     (if (<= KbT 1.7e-157)
       (/ NdChar (+ 2.0 (* mu (+ (/ 1.0 KbT) (/ (* mu 0.5) (* KbT KbT))))))
       (if (<= KbT 4.9e-45)
         (/
          NdChar
          (-
           (+
            (+ (/ EDonor KbT) 2.0)
            (* Vef (+ (/ 1.0 KbT) (/ mu (* Vef KbT)))))
           (/ Ec KbT)))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.35e+39) {
		tmp = t_0;
	} else if (KbT <= 1.7e-157) {
		tmp = NdChar / (2.0 + (mu * ((1.0 / KbT) + ((mu * 0.5) / (KbT * KbT)))));
	} else if (KbT <= 4.9e-45) {
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + (Vef * ((1.0 / KbT) + (mu / (Vef * KbT))))) - (Ec / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.35d+39)) then
        tmp = t_0
    else if (kbt <= 1.7d-157) then
        tmp = ndchar / (2.0d0 + (mu * ((1.0d0 / kbt) + ((mu * 0.5d0) / (kbt * kbt)))))
    else if (kbt <= 4.9d-45) then
        tmp = ndchar / ((((edonor / kbt) + 2.0d0) + (vef * ((1.0d0 / kbt) + (mu / (vef * kbt))))) - (ec / kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.35e+39) {
		tmp = t_0;
	} else if (KbT <= 1.7e-157) {
		tmp = NdChar / (2.0 + (mu * ((1.0 / KbT) + ((mu * 0.5) / (KbT * KbT)))));
	} else if (KbT <= 4.9e-45) {
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + (Vef * ((1.0 / KbT) + (mu / (Vef * KbT))))) - (Ec / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.35e+39:
		tmp = t_0
	elif KbT <= 1.7e-157:
		tmp = NdChar / (2.0 + (mu * ((1.0 / KbT) + ((mu * 0.5) / (KbT * KbT)))))
	elif KbT <= 4.9e-45:
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + (Vef * ((1.0 / KbT) + (mu / (Vef * KbT))))) - (Ec / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.35e+39)
		tmp = t_0;
	elseif (KbT <= 1.7e-157)
		tmp = Float64(NdChar / Float64(2.0 + Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Float64(mu * 0.5) / Float64(KbT * KbT))))));
	elseif (KbT <= 4.9e-45)
		tmp = Float64(NdChar / Float64(Float64(Float64(Float64(EDonor / KbT) + 2.0) + Float64(Vef * Float64(Float64(1.0 / KbT) + Float64(mu / Float64(Vef * KbT))))) - Float64(Ec / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.35e+39)
		tmp = t_0;
	elseif (KbT <= 1.7e-157)
		tmp = NdChar / (2.0 + (mu * ((1.0 / KbT) + ((mu * 0.5) / (KbT * KbT)))));
	elseif (KbT <= 4.9e-45)
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + (Vef * ((1.0 / KbT) + (mu / (Vef * KbT))))) - (Ec / KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.35e+39], t$95$0, If[LessEqual[KbT, 1.7e-157], N[(NdChar / N[(2.0 + N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(mu * 0.5), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.9e-45], N[(NdChar / N[(N[(N[(N[(EDonor / KbT), $MachinePrecision] + 2.0), $MachinePrecision] + N[(Vef * N[(N[(1.0 / KbT), $MachinePrecision] + N[(mu / N[(Vef * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.35000000000000002e39 or 4.8999999999999998e-45 < KbT

   1. Initial program 99.9%

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

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

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 46.6%

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

   7. Simplified 46.6%

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

   if -1.35000000000000002e39 < KbT < 1.69999999999999989e-157

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 32.3%

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

   10. Simplified 32.3%

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

   11. Taylor expanded in mu around 0

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

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

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. /-lowering-/.f64 25.9%

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

   13. Simplified 25.9%

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

   if 1.69999999999999989e-157 < KbT < 4.8999999999999998e-45

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in KbT around inf

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

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

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

      2. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 27.7%

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

   10. Simplified 27.7%

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

   11. Taylor expanded in Vef around inf

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

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

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

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

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

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

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 46.3%

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

   13. Simplified 46.3%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.35 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.7 \cdot 10^{-157}:\\ \;\;\;\;\frac{NdChar}{2 + mu \cdot \left(\frac{1}{KbT} + \frac{mu \cdot 0.5}{KbT \cdot KbT}\right)}\\ \mathbf{elif}\;KbT \leq 4.9 \cdot 10^{-45}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\frac{EDonor}{KbT} + 2\right) + Vef \cdot \left(\frac{1}{KbT} + \frac{mu}{Vef \cdot KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -7.2 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.1 \cdot 10^{-46}:\\ \;\;\;\;\frac{NdChar}{Vef \cdot \left(\frac{1}{KbT} + \frac{\left(\frac{EDonor}{KbT} + 2\right) + \frac{mu}{KbT}}{Vef}\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -7.2e+39)
     t_0
     (if (<= KbT 2.1e-46)
       (/
        NdChar
        (-
         (* Vef (+ (/ 1.0 KbT) (/ (+ (+ (/ EDonor KbT) 2.0) (/ mu KbT)) Vef)))
         (/ Ec KbT)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -7.2e+39) {
		tmp = t_0;
	} else if (KbT <= 2.1e-46) {
		tmp = NdChar / ((Vef * ((1.0 / KbT) + ((((EDonor / KbT) + 2.0) + (mu / KbT)) / Vef))) - (Ec / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-7.2d+39)) then
        tmp = t_0
    else if (kbt <= 2.1d-46) then
        tmp = ndchar / ((vef * ((1.0d0 / kbt) + ((((edonor / kbt) + 2.0d0) + (mu / kbt)) / vef))) - (ec / kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -7.2e+39) {
		tmp = t_0;
	} else if (KbT <= 2.1e-46) {
		tmp = NdChar / ((Vef * ((1.0 / KbT) + ((((EDonor / KbT) + 2.0) + (mu / KbT)) / Vef))) - (Ec / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -7.2e+39:
		tmp = t_0
	elif KbT <= 2.1e-46:
		tmp = NdChar / ((Vef * ((1.0 / KbT) + ((((EDonor / KbT) + 2.0) + (mu / KbT)) / Vef))) - (Ec / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -7.2e+39)
		tmp = t_0;
	elseif (KbT <= 2.1e-46)
		tmp = Float64(NdChar / Float64(Float64(Vef * Float64(Float64(1.0 / KbT) + Float64(Float64(Float64(Float64(EDonor / KbT) + 2.0) + Float64(mu / KbT)) / Vef))) - Float64(Ec / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -7.2e+39)
		tmp = t_0;
	elseif (KbT <= 2.1e-46)
		tmp = NdChar / ((Vef * ((1.0 / KbT) + ((((EDonor / KbT) + 2.0) + (mu / KbT)) / Vef))) - (Ec / KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -7.2e+39], t$95$0, If[LessEqual[KbT, 2.1e-46], N[(NdChar / N[(N[(Vef * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(N[(N[(EDonor / KbT), $MachinePrecision] + 2.0), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision] / Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -7.19999999999999969e39 or 2.09999999999999987e-46 < KbT

   1. Initial program 99.9%

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

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

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 46.6%

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

   7. Simplified 46.6%

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

   if -7.19999999999999969e39 < KbT < 2.09999999999999987e-46

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in KbT around inf

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

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

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

      2. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 22.4%

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

   10. Simplified 22.4%

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

   11. Taylor expanded in Vef around -inf

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

       1. associate-*r* N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(0 - Vef\right), \left(-1 \cdot \frac{2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)}{Vef} - \frac{1}{KbT}\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \left(-1 \cdot \frac{2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)}{Vef} - \frac{1}{KbT}\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \left(-1 \cdot \frac{2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)}{Vef} + \left(\mathsf{neg}\left(\frac{1}{KbT}\right)\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       7. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \left(-1 \cdot \frac{2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)}{Vef} + \frac{\mathsf{neg}\left(1\right)}{KbT}\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \left(-1 \cdot \frac{2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)}{Vef} + \frac{-1}{KbT}\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\left(-1 \cdot \frac{2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)}{Vef}\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)}{Vef}\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       11. distribute-neg-frac2 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\left(\frac{2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)}{\mathsf{neg}\left(Vef\right)}\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       12. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\left(\frac{2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)}{-1 \cdot Vef}\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(\frac{EDonor}{KbT} + \frac{mu}{KbT}\right)\right), \left(-1 \cdot Vef\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       14. associate-+r+ N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(2 + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}\right), \left(-1 \cdot Vef\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 + \frac{EDonor}{KbT}\right), \left(\frac{mu}{KbT}\right)\right), \left(-1 \cdot Vef\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{EDonor}{KbT}\right)\right), \left(\frac{mu}{KbT}\right)\right), \left(-1 \cdot Vef\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \left(\frac{mu}{KbT}\right)\right), \left(-1 \cdot Vef\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{/.f64}\left(mu, KbT\right)\right), \left(-1 \cdot Vef\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       19. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{/.f64}\left(mu, KbT\right)\right), \left(\mathsf{neg}\left(Vef\right)\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       20. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{/.f64}\left(mu, KbT\right)\right), \left(0 - Vef\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

       21. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, Vef\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(EDonor, KbT\right)\right), \mathsf{/.f64}\left(mu, KbT\right)\right), \mathsf{\_.f64}\left(0, Vef\right)\right), \left(\frac{-1}{KbT}\right)\right)\right), \mathsf{/.f64}\left(Ec, KbT\right)\right)\right) \]

   13. Simplified 27.5%

       \[\leadsto \frac{NdChar}{\color{blue}{\left(0 - Vef\right) \cdot \left(\frac{\left(2 + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}}{0 - Vef} + \frac{-1}{KbT}\right)} - \frac{Ec}{KbT}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -7.2 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.1 \cdot 10^{-46}:\\ \;\;\;\;\frac{NdChar}{Vef \cdot \left(\frac{1}{KbT} + \frac{\left(\frac{EDonor}{KbT} + 2\right) + \frac{mu}{KbT}}{Vef}\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.5% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -8 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\frac{EDonor}{KbT} + 2\right) + mu \cdot \left(\frac{1}{KbT} + \frac{Vef}{mu \cdot KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -8e+39)
     t_0
     (if (<= KbT 9.6e-5)
       (/
        NdChar
        (-
         (+ (+ (/ EDonor KbT) 2.0) (* mu (+ (/ 1.0 KbT) (/ Vef (* mu KbT)))))
         (/ Ec KbT)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -8e+39) {
		tmp = t_0;
	} else if (KbT <= 9.6e-5) {
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + (mu * ((1.0 / KbT) + (Vef / (mu * KbT))))) - (Ec / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-8d+39)) then
        tmp = t_0
    else if (kbt <= 9.6d-5) then
        tmp = ndchar / ((((edonor / kbt) + 2.0d0) + (mu * ((1.0d0 / kbt) + (vef / (mu * kbt))))) - (ec / kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -8e+39) {
		tmp = t_0;
	} else if (KbT <= 9.6e-5) {
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + (mu * ((1.0 / KbT) + (Vef / (mu * KbT))))) - (Ec / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -8e+39:
		tmp = t_0
	elif KbT <= 9.6e-5:
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + (mu * ((1.0 / KbT) + (Vef / (mu * KbT))))) - (Ec / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -8e+39)
		tmp = t_0;
	elseif (KbT <= 9.6e-5)
		tmp = Float64(NdChar / Float64(Float64(Float64(Float64(EDonor / KbT) + 2.0) + Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Vef / Float64(mu * KbT))))) - Float64(Ec / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -8e+39)
		tmp = t_0;
	elseif (KbT <= 9.6e-5)
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + (mu * ((1.0 / KbT) + (Vef / (mu * KbT))))) - (Ec / KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -8e+39], t$95$0, If[LessEqual[KbT, 9.6e-5], N[(NdChar / N[(N[(N[(N[(EDonor / KbT), $MachinePrecision] + 2.0), $MachinePrecision] + N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(Vef / N[(mu * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -7.99999999999999952e39 or 9.6000000000000002e-5 < KbT

   1. Initial program 99.9%

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

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

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 48.3%

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

   7. Simplified 48.3%

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

   if -7.99999999999999952e39 < KbT < 9.6000000000000002e-5

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 67.2%

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

   7. Simplified 67.2%

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

   8. Taylor expanded in KbT around inf

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

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

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

      2. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 21.5%

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

   10. Simplified 21.5%

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

   11. Taylor expanded in mu around inf

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

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

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

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

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

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

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

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

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

       5. *-lowering-*.f64 25.6%

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

   13. Simplified 25.6%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -8 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\frac{EDonor}{KbT} + 2\right) + mu \cdot \left(\frac{1}{KbT} + \frac{Vef}{mu \cdot KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -2.15 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.9 \cdot 10^{-80}:\\ \;\;\;\;\frac{NdChar}{2 + mu \cdot \left(\frac{1}{KbT} + \frac{mu \cdot 0.5}{KbT \cdot KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -2.15e+39)
     t_0
     (if (<= KbT 1.9e-80)
       (/ NdChar (+ 2.0 (* mu (+ (/ 1.0 KbT) (/ (* mu 0.5) (* KbT KbT))))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.15e+39) {
		tmp = t_0;
	} else if (KbT <= 1.9e-80) {
		tmp = NdChar / (2.0 + (mu * ((1.0 / KbT) + ((mu * 0.5) / (KbT * KbT)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-2.15d+39)) then
        tmp = t_0
    else if (kbt <= 1.9d-80) then
        tmp = ndchar / (2.0d0 + (mu * ((1.0d0 / kbt) + ((mu * 0.5d0) / (kbt * kbt)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.15e+39) {
		tmp = t_0;
	} else if (KbT <= 1.9e-80) {
		tmp = NdChar / (2.0 + (mu * ((1.0 / KbT) + ((mu * 0.5) / (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 <= -2.15e+39:
		tmp = t_0
	elif KbT <= 1.9e-80:
		tmp = NdChar / (2.0 + (mu * ((1.0 / KbT) + ((mu * 0.5) / (KbT * KbT)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -2.15e+39)
		tmp = t_0;
	elseif (KbT <= 1.9e-80)
		tmp = Float64(NdChar / Float64(2.0 + Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Float64(mu * 0.5) / 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 <= -2.15e+39)
		tmp = t_0;
	elseif (KbT <= 1.9e-80)
		tmp = NdChar / (2.0 + (mu * ((1.0 / KbT) + ((mu * 0.5) / (KbT * KbT)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -2.15e+39], t$95$0, If[LessEqual[KbT, 1.9e-80], N[(NdChar / N[(2.0 + N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(mu * 0.5), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -2.15e39 or 1.89999999999999983e-80 < KbT

   1. Initial program 99.9%

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

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

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 -2.15e39 < KbT < 1.89999999999999983e-80

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 31.1%

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

   10. Simplified 31.1%

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

   11. Taylor expanded in mu around 0

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

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

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. /-lowering-/.f64 24.3%

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

   13. Simplified 24.3%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.15 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.9 \cdot 10^{-80}:\\ \;\;\;\;\frac{NdChar}{2 + mu \cdot \left(\frac{1}{KbT} + \frac{mu \cdot 0.5}{KbT \cdot KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -9.5 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.45 \cdot 10^{-56}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -9.5e+38)
     t_0
     (if (<= KbT 1.45e-56) (/ NdChar (/ Vef KbT)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -9.5e+38) {
		tmp = t_0;
	} else if (KbT <= 1.45e-56) {
		tmp = NdChar / (Vef / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-9.5d+38)) then
        tmp = t_0
    else if (kbt <= 1.45d-56) then
        tmp = ndchar / (vef / kbt)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -9.5e+38) {
		tmp = t_0;
	} else if (KbT <= 1.45e-56) {
		tmp = NdChar / (Vef / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -9.5e+38:
		tmp = t_0
	elif KbT <= 1.45e-56:
		tmp = NdChar / (Vef / KbT)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -9.5e+38)
		tmp = t_0;
	elseif (KbT <= 1.45e-56)
		tmp = Float64(NdChar / Float64(Vef / KbT));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -9.5e+38)
		tmp = t_0;
	elseif (KbT <= 1.45e-56)
		tmp = NdChar / (Vef / KbT);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -9.5e+38], t$95$0, If[LessEqual[KbT, 1.45e-56], N[(NdChar / N[(Vef / KbT), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -9.4999999999999995e38 or 1.44999999999999996e-56 < KbT

   1. Initial program 99.9%

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

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

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 45.7%

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

   7. Simplified 45.7%

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

   if -9.4999999999999995e38 < KbT < 1.44999999999999996e-56

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in KbT around inf

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

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

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

      2. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 22.9%

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

   10. Simplified 22.9%

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

   11. Taylor expanded in Vef around inf

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

       1. /-lowering-/.f64 20.2%

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

   13. Simplified 20.2%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -9.5 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.45 \cdot 10^{-56}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 21.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq 2.8 \cdot 10^{+156}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -2.5e-157)
   (/ NdChar 2.0)
   (if (<= NdChar 2.8e+156) (* NaChar 0.5) (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -2.5e-157) {
		tmp = NdChar / 2.0;
	} else if (NdChar <= 2.8e+156) {
		tmp = NaChar * 0.5;
	} else {
		tmp = NdChar / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-2.5d-157)) then
        tmp = ndchar / 2.0d0
    else if (ndchar <= 2.8d+156) then
        tmp = nachar * 0.5d0
    else
        tmp = ndchar / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -2.5e-157) {
		tmp = NdChar / 2.0;
	} else if (NdChar <= 2.8e+156) {
		tmp = NaChar * 0.5;
	} else {
		tmp = NdChar / 2.0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -2.5e-157:
		tmp = NdChar / 2.0
	elif NdChar <= 2.8e+156:
		tmp = NaChar * 0.5
	else:
		tmp = NdChar / 2.0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -2.5e-157)
		tmp = Float64(NdChar / 2.0);
	elseif (NdChar <= 2.8e+156)
		tmp = Float64(NaChar * 0.5);
	else
		tmp = Float64(NdChar / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -2.5e-157)
		tmp = NdChar / 2.0;
	elseif (NdChar <= 2.8e+156)
		tmp = NaChar * 0.5;
	else
		tmp = NdChar / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -2.5e-157], N[(NdChar / 2.0), $MachinePrecision], If[LessEqual[NdChar, 2.8e+156], N[(NaChar * 0.5), $MachinePrecision], N[(NdChar / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.5000000000000001e-157 or 2.79999999999999988e156 < NdChar

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 72.8%

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

   7. Simplified 72.8%

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

   8. Taylor expanded in KbT around inf

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

   10. Simplified 24.6%

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

   if -2.5000000000000001e-157 < NdChar < 2.79999999999999988e156

   1. Initial program 100.0%

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

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

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Vef + \left(\left(EDonor + mu\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}} \]
   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 26.6%

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

   7. Simplified 26.6%

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

   8. Taylor expanded in NaChar around inf

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

      1. *-lowering-*.f64 25.4%

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

   10. Simplified 25.4%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq 2.8 \cdot 10^{+156}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 27.6% 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%

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

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

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

3. Simplified 100.0%

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

5. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

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

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

   3. +-lowering-+.f64 27.7%

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

7. Simplified 27.7%

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

8. Final simplification 27.7%

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

Alternative 19: 18.9% accurate, 76.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

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

3. Simplified 100.0%

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

5. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

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

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

   3. +-lowering-+.f64 27.7%

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

7. Simplified 27.7%

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

8. Taylor expanded in NaChar around inf

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

   1. *-lowering-*.f64 18.4%

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

10. Simplified 18.4%

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

11. Final simplification 18.4%

    \[\leadsto NaChar \cdot 0.5 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))))