Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 19.9s

Alternatives: 28

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = -2.6028817645623974e-15
  2. Ec = 3.743253668649183e-51
  3. Vef = -2.568384883876119e-94
  4. EDonor = 9.279894985489525e+262
  5. mu = -7.84849641222834e+159
  6. KbT = -2.7182112321085787e+110
  7. NaChar = -3.491547621850251e-16
  8. Ev = 254956426322.19507
  9. EAccept = 6.015257111880465e-166

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, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift--.f64 N/A

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

   4. distribute-frac-neg N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-/.f64 100.0

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

   10. lift--.f64 N/A

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

   11. lift--.f64 N/A

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

   12. associate--l- N/A

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

   13. lower--.f64 N/A

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

   14. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 51.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ t_1 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_1 \cdot t\_1}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;t\_0 \leq 10^{+93}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))))
        (t_1 (+ EDonor (+ Vef (- mu Ec)))))
   (if (<= t_0 -2e-99)
     (+ (/ NaChar (+ (exp (/ Vef KbT)) 1.0)) (* NdChar 0.5))
     (if (<= t_0 -2e-281)
       (/ NdChar (+ (exp (/ mu KbT)) 1.0))
       (if (<= t_0 0.0)
         (/
          NdChar
          (-
           2.0
           (/
            (fma -0.5 (/ (* t_1 t_1) KbT) (- (- (- Ec mu) Vef) EDonor))
            KbT)))
         (if (<= t_0 1e+93)
           (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
           (+ (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)) (* NdChar 0.5))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double t_1 = EDonor + (Vef + (mu - Ec));
	double tmp;
	if (t_0 <= -2e-99) {
		tmp = (NaChar / (exp((Vef / KbT)) + 1.0)) + (NdChar * 0.5);
	} else if (t_0 <= -2e-281) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else if (t_0 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_1 * t_1) / KbT), (((Ec - mu) - Vef) - EDonor)) / KbT));
	} else if (t_0 <= 1e+93) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = (NaChar / (exp((EAccept / KbT)) + 1.0)) + (NdChar * 0.5);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	t_1 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	tmp = 0.0
	if (t_0 <= -2e-99)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) + Float64(NdChar * 0.5));
	elseif (t_0 <= -2e-281)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_1 * t_1) / KbT), Float64(Float64(Float64(Ec - mu) - Vef) - EDonor)) / KbT)));
	elseif (t_0 <= 1e+93)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	else
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)) + Float64(NdChar * 0.5));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-99], N[(N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-281], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+93], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-99

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 99.9

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

      10. lift--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

      14. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 83.3

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.3

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

   10. Applied rewrites 58.3%

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

   if -2e-99 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-281

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in mu around inf

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

      1. lower-/.f64 53.5

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

   8. Applied rewrites 53.5%

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

   if -2e-281 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 88.9%

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

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.00000000000000004e93

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 46.3

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

   8. Applied rewrites 46.3%

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

   if 1.00000000000000004e93 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 10^{+93}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ t_2 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 10^{+93}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)) (* NdChar 0.5)))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))))
        (t_2 (+ EDonor (+ Vef (- mu Ec)))))
   (if (<= t_1 -1e-54)
     t_0
     (if (<= t_1 -2e-281)
       (/ NdChar (+ (exp (/ mu KbT)) 1.0))
       (if (<= t_1 0.0)
         (/
          NdChar
          (-
           2.0
           (/
            (fma -0.5 (/ (* t_2 t_2) KbT) (- (- (- Ec mu) Vef) EDonor))
            KbT)))
         (if (<= t_1 1e+93) (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (exp((EAccept / KbT)) + 1.0)) + (NdChar * 0.5);
	double t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double t_2 = EDonor + (Vef + (mu - Ec));
	double tmp;
	if (t_1 <= -1e-54) {
		tmp = t_0;
	} else if (t_1 <= -2e-281) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_2 * t_2) / KbT), (((Ec - mu) - Vef) - EDonor)) / KbT));
	} else if (t_1 <= 1e+93) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)) + Float64(NdChar * 0.5))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	t_2 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	tmp = 0.0
	if (t_1 <= -1e-54)
		tmp = t_0;
	elseif (t_1 <= -2e-281)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Float64(Float64(Float64(Ec - mu) - Vef) - EDonor)) / KbT)));
	elseif (t_1 <= 1e+93)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-54], t$95$0, If[LessEqual[t$95$1, -2e-281], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+93], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1e-54 or 1.00000000000000004e93 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.7

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

   8. Applied rewrites 53.7%

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

   if -1e-54 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-281

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in mu around inf

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

      1. lower-/.f64 45.5

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

   8. Applied rewrites 45.5%

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

   if -2e-281 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 88.9%

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

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.00000000000000004e93

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 46.3

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

   8. Applied rewrites 46.3%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-54}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 10^{+93}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ t_2 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_1 \cdot t\_1}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1 (+ EDonor (+ Vef (- mu Ec))))
        (t_2
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_2 -2e+121)
     t_0
     (if (<= t_2 -1e-291)
       (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
       (if (<= t_2 0.0)
         (/
          NdChar
          (-
           2.0
           (/
            (fma -0.5 (/ (* t_1 t_1) KbT) (- (- (- Ec mu) Vef) EDonor))
            KbT)))
         (if (<= t_2 5e-70) (/ 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 = 0.5 * (NdChar + NaChar);
	double t_1 = EDonor + (Vef + (mu - Ec));
	double t_2 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -2e+121) {
		tmp = t_0;
	} else if (t_2 <= -1e-291) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else if (t_2 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_1 * t_1) / KbT), (((Ec - mu) - Vef) - EDonor)) / KbT));
	} else if (t_2 <= 5e-70) {
		tmp = NaChar / (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(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	t_2 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -2e+121)
		tmp = t_0;
	elseif (t_2 <= -1e-291)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (t_2 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_1 * t_1) / KbT), Float64(Float64(Float64(Ec - mu) - Vef) - EDonor)) / KbT)));
	elseif (t_2 <= 5e-70)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return 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]}, Block[{t$95$1 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+121], t$95$0, If[LessEqual[t$95$2, -1e-291], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-70], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.00000000000000007e121 or 4.9999999999999998e-70 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 46.1

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

   5. Applied rewrites 46.1%

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

   if -2.00000000000000007e121 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999962e-292

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 99.9

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

      10. lift--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

      14. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 70.4

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 43.0

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

   10. Applied rewrites 43.0%

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

   if -9.99999999999999962e-292 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 90.6%

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

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999998e-70

   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. Add Preprocessing

   3. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 30.5

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

   8. Applied rewrites 30.5%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -2 \cdot 10^{+121}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ t_1 := t\_0 + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\frac{NdChar}{{\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-46}:\\ \;\;\;\;t\_0 + NdChar \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(NaChar, 0.5, \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))
        (t_1
         (+
          t_0
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_1 -1e+107)
     (+
      (/ NdChar (+ (pow (exp -1.0) (/ (- (- Ec Vef) (+ EDonor mu)) KbT)) 1.0))
      (/ NaChar 2.0))
     (if (<= t_1 -2e-46)
       (+ t_0 (* NdChar 0.5))
       (if (<= t_1 5e-30)
         (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
         (fma
          NaChar
          0.5
          (/ NdChar (+ (exp (/ (+ EDonor (+ Vef (- 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) {
	double t_0 = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	double t_1 = t_0 + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e+107) {
		tmp = (NdChar / (pow(exp(-1.0), (((Ec - Vef) - (EDonor + mu)) / KbT)) + 1.0)) + (NaChar / 2.0);
	} else if (t_1 <= -2e-46) {
		tmp = t_0 + (NdChar * 0.5);
	} else if (t_1 <= 5e-30) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = fma(NaChar, 0.5, (NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e+107)
		tmp = Float64(Float64(NdChar / Float64((exp(-1.0) ^ Float64(Float64(Float64(Ec - Vef) - Float64(EDonor + mu)) / KbT)) + 1.0)) + Float64(NaChar / 2.0));
	elseif (t_1 <= -2e-46)
		tmp = Float64(t_0 + Float64(NdChar * 0.5));
	elseif (t_1 <= 5e-30)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = fma(NaChar, 0.5, Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT)) + 1.0)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+107], N[(N[(NdChar / N[(N[Power[N[Exp[-1.0], $MachinePrecision], N[(N[(N[(Ec - Vef), $MachinePrecision] - N[(EDonor + mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-46], N[(t$95$0 + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-30], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar * 0.5 + N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999997e106

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 99.9

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

      10. lift--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

      14. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 89.5

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in Vef around 0

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

   10. Applied rewrites 80.9%

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

   if -9.9999999999999997e106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.00000000000000005e-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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -2.00000000000000005e-46 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999972e-30

   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. Add Preprocessing

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if 4.99999999999999972e-30 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in EAccept around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 71.7

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

   8. Applied rewrites 71.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\frac{NdChar}{{\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)} + 1} + \frac{NaChar}{2}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -2 \cdot 10^{-46}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(NaChar, 0.5, \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(NaChar, 0.5, \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\right)\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ t_2 := t\_1 + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-46}:\\ \;\;\;\;t\_1 + NdChar \cdot 0.5\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (fma
          NaChar
          0.5
          (/ NdChar (+ (exp (/ (+ EDonor (+ Vef (- mu Ec))) KbT)) 1.0))))
        (t_1 (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))
        (t_2
         (+
          t_1
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_2 -1e+107)
     t_0
     (if (<= t_2 -2e-46)
       (+ t_1 (* NdChar 0.5))
       (if (<= t_2 5e-30)
         (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fma(NaChar, 0.5, (NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0)));
	double t_1 = NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0);
	double t_2 = t_1 + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -1e+107) {
		tmp = t_0;
	} else if (t_2 <= -2e-46) {
		tmp = t_1 + (NdChar * 0.5);
	} else if (t_2 <= 5e-30) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = fma(NaChar, 0.5, Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT)) + 1.0)))
	t_1 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0))
	t_2 = Float64(t_1 + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -1e+107)
		tmp = t_0;
	elseif (t_2 <= -2e-46)
		tmp = Float64(t_1 + Float64(NdChar * 0.5));
	elseif (t_2 <= 5e-30)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar * 0.5 + N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+107], t$95$0, If[LessEqual[t$95$2, -2e-46], N[(t$95$1 + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-30], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999997e106 or 4.99999999999999972e-30 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in EAccept around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 75.3

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

   8. Applied rewrites 75.3%

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

   if -9.9999999999999997e106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.00000000000000005e-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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -2.00000000000000005e-46 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999972e-30

   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. Add Preprocessing

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 75.3

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

   5. Applied rewrites 75.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(NaChar, 0.5, \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -2 \cdot 10^{-46}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(NaChar, 0.5, \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_2 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (+ Vef (- mu Ec))))
        (t_1 (* 0.5 (+ NdChar NaChar)))
        (t_2
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_2 -1e-239)
     t_1
     (if (<= t_2 0.0)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- (- Ec mu) Vef) EDonor)) KbT)))
       (if (<= t_2 5e-70) (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)) t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef + (mu - Ec));
	double t_1 = 0.5 * (NdChar + NaChar);
	double t_2 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -1e-239) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), (((Ec - mu) - Vef) - EDonor)) / KbT));
	} else if (t_2 <= 5e-70) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	t_2 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -1e-239)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Float64(Ec - mu) - Vef) - EDonor)) / KbT)));
	elseif (t_2 <= 5e-70)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-239], t$95$1, If[LessEqual[t$95$2, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-70], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.0000000000000001e-239 or 4.9999999999999998e-70 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 42.6

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

   5. Applied rewrites 42.6%

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

   if -1.0000000000000001e-239 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 96.6

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 81.2%

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

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999998e-70

   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. Add Preprocessing

   3. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 30.5

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

   8. Applied rewrites 30.5%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-239}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-252}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(Vef + mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{-79}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          t_0)))
   (if (<= t_1 -5e-252)
     (+
      (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
      (/ NdChar (+ (exp (/ (+ EDonor (+ Vef mu)) KbT)) 1.0)))
     (if (<= t_1 1e-79)
       (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
       (+ t_0 (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	double t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + t_0;
	double tmp;
	if (t_1 <= -5e-252) {
		tmp = (NaChar / (exp((Vef / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (Vef + mu)) / KbT)) + 1.0));
	} else if (t_1 <= 1e-79) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0 + (NaChar / (exp((EAccept / KbT)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)
    t_1 = (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)) + t_0
    if (t_1 <= (-5d-252)) then
        tmp = (nachar / (exp((vef / kbt)) + 1.0d0)) + (ndchar / (exp(((edonor + (vef + mu)) / kbt)) + 1.0d0))
    else if (t_1 <= 1d-79) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else
        tmp = t_0 + (nachar / (exp((eaccept / kbt)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	double t_1 = (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + t_0;
	double tmp;
	if (t_1 <= -5e-252) {
		tmp = (NaChar / (Math.exp((Vef / KbT)) + 1.0)) + (NdChar / (Math.exp(((EDonor + (Vef + mu)) / KbT)) + 1.0));
	} else if (t_1 <= 1e-79) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0 + (NaChar / (Math.exp((EAccept / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)
	t_1 = (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + t_0
	tmp = 0
	if t_1 <= -5e-252:
		tmp = (NaChar / (math.exp((Vef / KbT)) + 1.0)) + (NdChar / (math.exp(((EDonor + (Vef + mu)) / KbT)) + 1.0))
	elif t_1 <= 1e-79:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	else:
		tmp = t_0 + (NaChar / (math.exp((EAccept / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + t_0)
	tmp = 0.0
	if (t_1 <= -5e-252)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + mu)) / KbT)) + 1.0)));
	elseif (t_1 <= 1e-79)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + t_0;
	tmp = 0.0;
	if (t_1 <= -5e-252)
		tmp = (NaChar / (exp((Vef / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (Vef + mu)) / KbT)) + 1.0));
	elseif (t_1 <= 1e-79)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	else
		tmp = t_0 + (NaChar / (exp((EAccept / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-252], N[(N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-79], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000008e-252

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 99.9

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

      10. lift--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

      14. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in Ec around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 77.0

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

   10. Applied rewrites 77.0%

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

   if -5.00000000000000008e-252 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1e-79

   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. Add Preprocessing

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if 1e-79 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -5 \cdot 10^{-252}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(Vef + mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 10^{-79}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(Vef + mu\right)}{KbT}} + 1}\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-252}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ EDonor (+ Vef mu)) KbT)) 1.0))))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_1 -5e-252)
     t_0
     (if (<= t_1 5e-30)
       (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (exp((Vef / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (Vef + mu)) / KbT)) + 1.0));
	double t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-252) {
		tmp = t_0;
	} else if (t_1 <= 5e-30) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar / (exp((vef / kbt)) + 1.0d0)) + (ndchar / (exp(((edonor + (vef + mu)) / kbt)) + 1.0d0))
    t_1 = (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)) + (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0))
    if (t_1 <= (-5d-252)) then
        tmp = t_0
    else if (t_1 <= 5d-30) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (Math.exp((Vef / KbT)) + 1.0)) + (NdChar / (Math.exp(((EDonor + (Vef + mu)) / KbT)) + 1.0));
	double t_1 = (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-252) {
		tmp = t_0;
	} else if (t_1 <= 5e-30) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (math.exp((Vef / KbT)) + 1.0)) + (NdChar / (math.exp(((EDonor + (Vef + mu)) / KbT)) + 1.0))
	t_1 = (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -5e-252:
		tmp = t_0
	elif t_1 <= 5e-30:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + mu)) / KbT)) + 1.0)))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e-252)
		tmp = t_0;
	elseif (t_1 <= 5e-30)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (exp((Vef / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (Vef + mu)) / KbT)) + 1.0));
	t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -5e-252)
		tmp = t_0;
	elseif (t_1 <= 5e-30)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-252], t$95$0, If[LessEqual[t$95$1, 5e-30], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000008e-252 or 4.99999999999999972e-30 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 100.0

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

      10. lift--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

      14. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 79.7

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

   7. Applied rewrites 79.7%

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

   8. Taylor expanded in Ec around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 75.0

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

   10. Applied rewrites 75.0%

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

   if -5.00000000000000008e-252 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999972e-30

   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. Add Preprocessing

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -5 \cdot 10^{-252}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(Vef + mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(Vef + mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(NaChar, 0.5, \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\right)\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (fma
          NaChar
          0.5
          (/ NdChar (+ (exp (/ (+ EDonor (+ Vef (- mu Ec))) KbT)) 1.0))))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_1 -1e+107)
     t_0
     (if (<= t_1 5e-30)
       (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fma(NaChar, 0.5, (NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0)));
	double t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e+107) {
		tmp = t_0;
	} else if (t_1 <= 5e-30) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = fma(NaChar, 0.5, Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT)) + 1.0)))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e+107)
		tmp = t_0;
	elseif (t_1 <= 5e-30)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar * 0.5 + N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+107], t$95$0, If[LessEqual[t$95$1, 5e-30], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999997e106 or 4.99999999999999972e-30 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in EAccept around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 75.3

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

   8. Applied rewrites 75.3%

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

   if -9.9999999999999997e106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999972e-30

   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. Add Preprocessing

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 73.5

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

   5. Applied rewrites 73.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(NaChar, 0.5, \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(NaChar, 0.5, \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_2 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-148}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (+ Vef (- mu Ec))))
        (t_1 (* 0.5 (+ NdChar NaChar)))
        (t_2
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_2 -1e-239)
     t_1
     (if (<= t_2 1e-148)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- (- Ec mu) Vef) EDonor)) KbT)))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef + (mu - Ec));
	double t_1 = 0.5 * (NdChar + NaChar);
	double t_2 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -1e-239) {
		tmp = t_1;
	} else if (t_2 <= 1e-148) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), (((Ec - mu) - Vef) - EDonor)) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	t_2 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -1e-239)
		tmp = t_1;
	elseif (t_2 <= 1e-148)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Float64(Ec - mu) - Vef) - EDonor)) / KbT)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-239], t$95$1, If[LessEqual[t$95$2, 1e-148], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.0000000000000001e-239 or 9.99999999999999936e-149 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 41.5

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

   5. Applied rewrites 41.5%

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

   if -1.0000000000000001e-239 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.99999999999999936e-149

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 62.0%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-239}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 10^{-148}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - mu\right) - Vef\right) - EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef + mu\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_2 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-148}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(-EDonor\right) - \left(Vef + mu\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (+ Vef mu)))
        (t_1 (* 0.5 (+ NdChar NaChar)))
        (t_2
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_2 -1e-239)
     t_1
     (if (<= t_2 1e-148)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- EDonor) (+ Vef mu))) KbT)))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef + mu);
	double t_1 = 0.5 * (NdChar + NaChar);
	double t_2 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -1e-239) {
		tmp = t_1;
	} else if (t_2 <= 1e-148) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), (-EDonor - (Vef + mu))) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef + mu))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	t_2 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -1e-239)
		tmp = t_1;
	elseif (t_2 <= 1e-148)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(-EDonor) - Float64(Vef + mu))) / KbT)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-239], t$95$1, If[LessEqual[t$95$2, 1e-148], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[((-EDonor) - N[(Vef + mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.0000000000000001e-239 or 9.99999999999999936e-149 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 41.5

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

   5. Applied rewrites 41.5%

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

   if -1.0000000000000001e-239 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.99999999999999936e-149

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 62.0%

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

   9. Taylor expanded in Ec around 0

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

       1. sub-neg N/A

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

       2. lower-fma.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(EDonor + \left(Vef + mu\right)\right)}^{2}}{KbT}, \mathsf{neg}\left(\left(EDonor + \left(Vef + mu\right)\right)\right)\right)}}{KbT}} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{\left(EDonor + \left(Vef + mu\right)\right)}^{2}}{KbT}}, \mathsf{neg}\left(\left(EDonor + \left(Vef + mu\right)\right)\right)\right)}{KbT}} \]

       4. unpow2 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(EDonor + \left(Vef + mu\right)\right) \cdot \left(EDonor + \left(Vef + mu\right)\right)}}{KbT}, \mathsf{neg}\left(\left(EDonor + \left(Vef + mu\right)\right)\right)\right)}{KbT}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(EDonor + \left(Vef + mu\right)\right) \cdot \left(EDonor + \left(Vef + mu\right)\right)}}{KbT}, \mathsf{neg}\left(\left(EDonor + \left(Vef + mu\right)\right)\right)\right)}{KbT}} \]

       6. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(EDonor + \left(Vef + mu\right)\right)} \cdot \left(EDonor + \left(Vef + mu\right)\right)}{KbT}, \mathsf{neg}\left(\left(EDonor + \left(Vef + mu\right)\right)\right)\right)}{KbT}} \]

       7. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(EDonor + \color{blue}{\left(Vef + mu\right)}\right) \cdot \left(EDonor + \left(Vef + mu\right)\right)}{KbT}, \mathsf{neg}\left(\left(EDonor + \left(Vef + mu\right)\right)\right)\right)}{KbT}} \]

       8. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(EDonor + \left(Vef + mu\right)\right) \cdot \color{blue}{\left(EDonor + \left(Vef + mu\right)\right)}}{KbT}, \mathsf{neg}\left(\left(EDonor + \left(Vef + mu\right)\right)\right)\right)}{KbT}} \]

       9. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(EDonor + \left(Vef + mu\right)\right) \cdot \left(EDonor + \color{blue}{\left(Vef + mu\right)}\right)}{KbT}, \mathsf{neg}\left(\left(EDonor + \left(Vef + mu\right)\right)\right)\right)}{KbT}} \]

       10. lower-neg.f64 N/A

           \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(EDonor + \left(Vef + mu\right)\right) \cdot \left(EDonor + \left(Vef + mu\right)\right)}{KbT}, \color{blue}{\mathsf{neg}\left(\left(EDonor + \left(Vef + mu\right)\right)\right)}\right)}{KbT}} \]

       11. lower-+.f64 N/A

           \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(EDonor + \left(Vef + mu\right)\right) \cdot \left(EDonor + \left(Vef + mu\right)\right)}{KbT}, \mathsf{neg}\left(\color{blue}{\left(EDonor + \left(Vef + mu\right)\right)}\right)\right)}{KbT}} \]

       12. lower-+.f64 59.5

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

   11. Applied rewrites 59.5%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-239}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 10^{-148}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + mu\right)\right) \cdot \left(EDonor + \left(Vef + mu\right)\right)}{KbT}, \left(-EDonor\right) - \left(Vef + mu\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-239}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_1 -1e-239)
     t_0
     (if (<= t_1 0.0)
       (/ NdChar (+ 2.0 (/ (- (+ EDonor (+ Vef mu)) 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 t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-239) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 + (((EDonor + (Vef + mu)) - 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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)) + (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0))
    if (t_1 <= (-1d-239)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = ndchar / (2.0d0 + (((edonor + (vef + mu)) - 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 t_1 = (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-239) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 + (((EDonor + (Vef + mu)) - 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)
	t_1 = (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -1e-239:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = NdChar / (2.0 + (((EDonor + (Vef + mu)) - 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))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-239)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(EDonor + Float64(Vef + mu)) - 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);
	t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -1e-239)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = NdChar / (2.0 + (((EDonor + (Vef + mu)) - 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]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-239], t$95$0, If[LessEqual[t$95$1, 0.0], N[(NdChar / N[(2.0 + N[(N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.0000000000000001e-239 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 38.5

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

   5. Applied rewrites 38.5%

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

   if -1.0000000000000001e-239 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 96.6

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in KbT around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-+.f64 58.8

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

   11. Applied rewrites 58.8%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-239}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-291}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Vef \cdot Vef}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_1 -1e-291)
     t_0
     (if (<= t_1 0.0) (/ (* 2.0 (* NdChar (* KbT KbT))) (* Vef Vef)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-291) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)) + (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0))
    if (t_1 <= (-1d-291)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (2.0d0 * (ndchar * (kbt * kbt))) / (vef * vef)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-291) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -1e-291:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-291)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(2.0 * Float64(NdChar * Float64(KbT * KbT))) / Float64(Vef * Vef));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -1e-291)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-291], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(2.0 * N[(NdChar * N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Vef * Vef), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999962e-292 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.5

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

   5. Applied rewrites 37.5%

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

   if -9.99999999999999962e-292 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 90.6%

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

   9. Taylor expanded in Vef around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-291}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Vef \cdot Vef}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-276}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Ec \cdot Ec}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_1 -1e-276)
     t_0
     (if (<= t_1 0.0) (/ (* 2.0 (* NdChar (* KbT KbT))) (* Ec Ec)) 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 t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-276) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)) + (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0))
    if (t_1 <= (-1d-276)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (2.0d0 * (ndchar * (kbt * kbt))) / (ec * ec)
    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 t_1 = (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-276) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -1e-276:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec)
	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))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-276)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(2.0 * Float64(NdChar * Float64(KbT * KbT))) / Float64(Ec * Ec));
	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);
	t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -1e-276)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec);
	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]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-276], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(2.0 * N[(NdChar * N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Ec * Ec), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1e-276 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.9

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

   5. Applied rewrites 37.9%

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

   if -1e-276 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 87.2%

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

   9. Taylor expanded in Ec around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 56.1

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

   11. Applied rewrites 56.1%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Ec \cdot Ec}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-291}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{EDonor \cdot EDonor}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_1 -1e-291)
     t_0
     (if (<= t_1 0.0)
       (/ (* 2.0 (* NdChar (* KbT KbT))) (* EDonor EDonor))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-291) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (EDonor * EDonor);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)) + (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0))
    if (t_1 <= (-1d-291)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (2.0d0 * (ndchar * (kbt * kbt))) / (edonor * edonor)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-291) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (EDonor * EDonor);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -1e-291:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (EDonor * EDonor)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-291)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(2.0 * Float64(NdChar * Float64(KbT * KbT))) / Float64(EDonor * EDonor));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -1e-291)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (EDonor * EDonor);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-291], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(2.0 * N[(NdChar * N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(EDonor * EDonor), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999962e-292 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.5

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

   5. Applied rewrites 37.5%

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

   if -9.99999999999999962e-292 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 90.6%

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

   9. Taylor expanded in EDonor around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 40.0

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

   11. Applied rewrites 40.0%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-291}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{EDonor \cdot EDonor}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 32.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-276}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{-0.5 \cdot \left(NdChar \cdot NdChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))))
   (if (<= t_1 -1e-276)
     t_0
     (if (<= t_1 0.0) (/ (* -0.5 (* NdChar NdChar)) (- NaChar NdChar)) 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 t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-276) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0)) + (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0))
    if (t_1 <= (-1d-276)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = ((-0.5d0) * (ndchar * ndchar)) / (nachar - ndchar)
    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 t_1 = (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-276) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -1e-276:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar)
	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))
	t_1 = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-276)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-0.5 * Float64(NdChar * NdChar)) / Float64(NaChar - NdChar));
	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);
	t_1 = (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -1e-276)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar);
	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]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-276], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(-0.5 * N[(NdChar * NdChar), $MachinePrecision]), $MachinePrecision] / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1e-276 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) 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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.9

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

   5. Applied rewrites 37.9%

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

   if -1e-276 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   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. Add Preprocessing

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 4.4

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in NdChar around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 29.0

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

   10. Applied rewrites 29.0%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq -1 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{-0.5 \cdot \left(NdChar \cdot NdChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 19: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 20: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;EAccept \leq 5.5 \cdot 10^{+191}:\\ \;\;\;\;t\_0 + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))))
   (if (<= EAccept 5.5e+191)
     (+ t_0 (/ NaChar (+ (exp (/ Vef KbT)) 1.0)))
     (+ t_0 (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	double tmp;
	if (EAccept <= 5.5e+191) {
		tmp = t_0 + (NaChar / (exp((Vef / KbT)) + 1.0));
	} else {
		tmp = t_0 + (NaChar / (exp((EAccept / KbT)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)
    if (eaccept <= 5.5d+191) then
        tmp = t_0 + (nachar / (exp((vef / kbt)) + 1.0d0))
    else
        tmp = t_0 + (nachar / (exp((eaccept / kbt)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	double tmp;
	if (EAccept <= 5.5e+191) {
		tmp = t_0 + (NaChar / (Math.exp((Vef / KbT)) + 1.0));
	} else {
		tmp = t_0 + (NaChar / (Math.exp((EAccept / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)
	tmp = 0
	if EAccept <= 5.5e+191:
		tmp = t_0 + (NaChar / (math.exp((Vef / KbT)) + 1.0))
	else:
		tmp = t_0 + (NaChar / (math.exp((EAccept / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0))
	tmp = 0.0
	if (EAccept <= 5.5e+191)
		tmp = Float64(t_0 + Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	tmp = 0.0;
	if (EAccept <= 5.5e+191)
		tmp = t_0 + (NaChar / (exp((Vef / KbT)) + 1.0));
	else
		tmp = t_0 + (NaChar / (exp((EAccept / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EAccept, 5.5e+191], N[(t$95$0 + N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if EAccept < 5.5000000000000002e191

   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. Add Preprocessing

   3. Taylor expanded in Vef around inf

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

      1. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if 5.5000000000000002e191 < EAccept

   1. Initial program 100.0%

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

   3. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 5.5 \cdot 10^{+191}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 68.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{+159}:\\ \;\;\;\;NdChar \cdot \frac{-1}{-1 - e^{\frac{Vef + \left(EDonor + \left(mu - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -2.4e+159)
   (* NdChar (/ -1.0 (- -1.0 (exp (/ (+ Vef (+ EDonor (- mu Ec))) KbT)))))
   (if (<= NdChar 2.8e+32)
     (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
     (/ NdChar (+ (exp (/ (+ EDonor (+ Vef (- 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) {
	double tmp;
	if (NdChar <= -2.4e+159) {
		tmp = NdChar * (-1.0 / (-1.0 - exp(((Vef + (EDonor + (mu - Ec))) / KbT))));
	} else if (NdChar <= 2.8e+32) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-2.4d+159)) then
        tmp = ndchar * ((-1.0d0) / ((-1.0d0) - exp(((vef + (edonor + (mu - ec))) / kbt))))
    else if (ndchar <= 2.8d+32) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else
        tmp = ndchar / (exp(((edonor + (vef + (mu - ec))) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -2.4e+159) {
		tmp = NdChar * (-1.0 / (-1.0 - Math.exp(((Vef + (EDonor + (mu - Ec))) / KbT))));
	} else if (NdChar <= 2.8e+32) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (Math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -2.4e+159:
		tmp = NdChar * (-1.0 / (-1.0 - math.exp(((Vef + (EDonor + (mu - Ec))) / KbT))))
	elif NdChar <= 2.8e+32:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	else:
		tmp = NdChar / (math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -2.4e+159)
		tmp = Float64(NdChar * Float64(-1.0 / Float64(-1.0 - exp(Float64(Float64(Vef + Float64(EDonor + Float64(mu - Ec))) / KbT)))));
	elseif (NdChar <= 2.8e+32)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -2.4e+159)
		tmp = NdChar * (-1.0 / (-1.0 - exp(((Vef + (EDonor + (mu - Ec))) / KbT))));
	elseif (NdChar <= 2.8e+32)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	else
		tmp = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -2.4e+159], N[(NdChar * N[(-1.0 / N[(-1.0 - N[Exp[N[(N[(Vef + N[(EDonor + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.8e+32], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -2.4e159

   1. Initial program 99.8%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 80.6%

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

   if -2.4e159 < NdChar < 2.8e32

   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. Add Preprocessing

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if 2.8e32 < 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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{+159}:\\ \;\;\;\;NdChar \cdot \frac{-1}{-1 - e^{\frac{Vef + \left(EDonor + \left(mu - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 68.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (+ EDonor (+ Vef (- mu Ec))) KbT)) 1.0))))
   (if (<= NdChar -2.4e+159)
     t_0
     (if (<= NdChar 2.8e+32)
       (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	double tmp;
	if (NdChar <= -2.4e+159) {
		tmp = t_0;
	} else if (NdChar <= 2.8e+32) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp(((edonor + (vef + (mu - ec))) / kbt)) + 1.0d0)
    if (ndchar <= (-2.4d+159)) then
        tmp = t_0
    else if (ndchar <= 2.8d+32) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	double tmp;
	if (NdChar <= -2.4e+159) {
		tmp = t_0;
	} else if (NdChar <= 2.8e+32) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0)
	tmp = 0
	if NdChar <= -2.4e+159:
		tmp = t_0
	elif NdChar <= 2.8e+32:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT)) + 1.0))
	tmp = 0.0
	if (NdChar <= -2.4e+159)
		tmp = t_0;
	elseif (NdChar <= 2.8e+32)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	tmp = 0.0;
	if (NdChar <= -2.4e+159)
		tmp = t_0;
	elseif (NdChar <= 2.8e+32)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -2.4e+159], t$95$0, If[LessEqual[NdChar, 2.8e+32], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.4e159 or 2.8e32 < NdChar

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -2.4e159 < NdChar < 2.8e32

   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. Add Preprocessing

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept + \left(\left(Ev + Vef\right) - mu\right)}}{KbT}}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept + \left(\left(Ev + Vef\right) - mu\right)}}{KbT}}} \]

      7. sub-neg N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)}}{KbT}}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)}}{KbT}}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef + \color{blue}{-1 \cdot mu}\right)\right)}{KbT}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(Ev + \left(Vef + -1 \cdot mu\right)\right)}}{KbT}}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef + \color{blue}{\left(\mathsf{neg}\left(mu\right)\right)}\right)\right)}{KbT}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \color{blue}{\left(Vef - mu\right)}\right)}{KbT}}} \]

      13. lower--.f64 71.0

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \color{blue}{\left(Vef - mu\right)}\right)}{KbT}}} \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{+159}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -3.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 5.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -3.6e+159)
   (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
   (if (<= NdChar 5.8e+41)
     (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
     (/ NdChar (+ (exp (/ Ec (- KbT))) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -3.6e+159) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else if (NdChar <= 5.8e+41) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-3.6d+159)) then
        tmp = ndchar / (exp((vef / kbt)) + 1.0d0)
    else if (ndchar <= 5.8d+41) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else
        tmp = ndchar / (exp((ec / -kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -3.6e+159) {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	} else if (NdChar <= 5.8e+41) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (Math.exp((Ec / -KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -3.6e+159:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	elif NdChar <= 5.8e+41:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	else:
		tmp = NdChar / (math.exp((Ec / -KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -3.6e+159)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (NdChar <= 5.8e+41)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -3.6e+159)
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	elseif (NdChar <= 5.8e+41)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	else
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -3.6e+159], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 5.8e+41], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -3.60000000000000037e159

   1. Initial program 99.8%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      7. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + -1 \cdot Ec\right)\right)}}{KbT}}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)\right)}{KbT}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

      13. lower--.f64 80.6

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}} \]

   6. Taylor expanded in Vef around inf

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 66.9

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   8. Applied rewrites 66.9%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if -3.60000000000000037e159 < NdChar < 5.79999999999999977e41

   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. Add Preprocessing

   3. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept + \left(\left(Ev + Vef\right) - mu\right)}}{KbT}}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept + \left(\left(Ev + Vef\right) - mu\right)}}{KbT}}} \]

      7. sub-neg N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)}}{KbT}}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)}}{KbT}}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef + \color{blue}{-1 \cdot mu}\right)\right)}{KbT}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(Ev + \left(Vef + -1 \cdot mu\right)\right)}}{KbT}}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef + \color{blue}{\left(\mathsf{neg}\left(mu\right)\right)}\right)\right)}{KbT}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \color{blue}{\left(Vef - mu\right)}\right)}{KbT}}} \]

      13. lower--.f64 71.0

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \color{blue}{\left(Vef - mu\right)}\right)}{KbT}}} \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}} \]

   if 5.79999999999999977e41 < 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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      7. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + -1 \cdot Ec\right)\right)}}{KbT}}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)\right)}{KbT}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

      13. lower--.f64 77.8

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

   5. Applied rewrites 77.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}} \]

   6. Taylor expanded in Ec around inf

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{-1 \cdot \frac{Ec}{KbT}}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{Ec}{KbT}\right)}}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{Ec}{KbT}\right)}}} \]

      3. lower-/.f64 56.5

         \[\leadsto \frac{NdChar}{1 + e^{-\color{blue}{\frac{Ec}{KbT}}}} \]

   8. Applied rewrites 56.5%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{-\frac{Ec}{KbT}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -3.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 5.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 43.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;NdChar \leq -9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{elif}\;NdChar \leq 8200:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0)))
   (if (<= NdChar -9.5e+112)
     (/ NdChar t_0)
     (if (<= NdChar 8200.0)
       (/ 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 <= -9.5e+112) {
		tmp = NdChar / t_0;
	} else if (NdChar <= 8200.0) {
		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 <= (-9.5d+112)) then
        tmp = ndchar / t_0
    else if (ndchar <= 8200.0d0) 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 <= -9.5e+112) {
		tmp = NdChar / t_0;
	} else if (NdChar <= 8200.0) {
		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 <= -9.5e+112:
		tmp = NdChar / t_0
	elif NdChar <= 8200.0:
		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 <= -9.5e+112)
		tmp = Float64(NdChar / t_0);
	elseif (NdChar <= 8200.0)
		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 <= -9.5e+112)
		tmp = NdChar / t_0;
	elseif (NdChar <= 8200.0)
		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, -9.5e+112], N[(NdChar / t$95$0), $MachinePrecision], If[LessEqual[NdChar, 8200.0], 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 < -9.5000000000000008e112

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      7. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + -1 \cdot Ec\right)\right)}}{KbT}}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)\right)}{KbT}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

      13. lower--.f64 75.0

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}} \]

   6. Taylor expanded in Vef around inf

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 62.7

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   8. Applied rewrites 62.7%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if -9.5000000000000008e112 < NdChar < 8200

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\color{blue}{\left(Ec - Vef\right)} - EDonor\right) - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\color{blue}{\left(\left(Ec - Vef\right) - EDonor\right)} - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      4. distribute-frac-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{-1 \cdot \frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      6. exp-prod N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + {\color{blue}{\left(e^{-1}\right)}}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\color{blue}{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\left(\left(Ec - Vef\right) - EDonor\right) - mu}}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\left(\left(Ec - Vef\right) - EDonor\right)} - mu}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      12. associate--l- N/A

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\left(Ec - Vef\right) - \left(EDonor + mu\right)}}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\left(Ec - Vef\right) - \left(EDonor + mu\right)}}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      14. lower-+.f64 100.0

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \color{blue}{\left(EDonor + mu\right)}}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   5. Taylor expanded in Vef around inf

      \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 67.4

         \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   7. Applied rewrites 67.4%

      \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   8. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{Vef}{KbT}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{Vef}{KbT}}}} \]

      4. lower-/.f64 49.3

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   10. Applied rewrites 49.3%

       \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}} \]

   if 8200 < 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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      7. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + -1 \cdot Ec\right)\right)}}{KbT}}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)\right)}{KbT}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

      13. lower--.f64 75.2

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}} \]

   6. Taylor expanded in EDonor around inf

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 55.8

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]

   8. Applied rewrites 55.8%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 8200:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 42.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -1.45 \cdot 10^{-107}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -1.45e-107)
   (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
   (if (<= NaChar 1.6e+57)
     (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
     (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.45e-107) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else if (NaChar <= 1.6e+57) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-1.45d-107)) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else if (nachar <= 1.6d+57) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.45e-107) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else if (NaChar <= 1.6e+57) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -1.45e-107:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	elif NaChar <= 1.6e+57:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -1.45e-107)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (NaChar <= 1.6e+57)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -1.45e-107)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	elseif (NaChar <= 1.6e+57)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	else
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -1.45e-107], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.6e+57], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -1.4499999999999999e-107

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\color{blue}{\left(Ec - Vef\right)} - EDonor\right) - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\color{blue}{\left(\left(Ec - Vef\right) - EDonor\right)} - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\color{blue}{\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      4. distribute-frac-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{-1 \cdot \frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      6. exp-prod N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + {\color{blue}{\left(e^{-1}\right)}}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\color{blue}{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\left(\left(Ec - Vef\right) - EDonor\right) - mu}}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\left(\left(Ec - Vef\right) - EDonor\right)} - mu}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      12. associate--l- N/A

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\left(Ec - Vef\right) - \left(EDonor + mu\right)}}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\color{blue}{\left(Ec - Vef\right) - \left(EDonor + mu\right)}}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(\mathsf{neg}\left(mu\right)\right)}{KbT}}} \]

      14. lower-+.f64 100.0

          \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \color{blue}{\left(EDonor + mu\right)}}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   5. Taylor expanded in Vef around inf

      \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 76.7

         \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   7. Applied rewrites 76.7%

      \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\left(\frac{\left(Ec - Vef\right) - \left(EDonor + mu\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   8. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{Vef}{KbT}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{Vef}{KbT}}}} \]

      4. lower-/.f64 59.3

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   10. Applied rewrites 59.3%

       \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}} \]

   if -1.4499999999999999e-107 < NaChar < 1.60000000000000015e57

   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. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      7. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + -1 \cdot Ec\right)\right)}}{KbT}}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)\right)}{KbT}}} \]

      12. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

      13. lower--.f64 74.1

          \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}}} \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}} \]

   6. Taylor expanded in EDonor around inf

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 50.0

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]

   if 1.60000000000000015e57 < 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. Add Preprocessing

   3. Taylor expanded in EAccept around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 58.9

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{EAccept}{KbT}}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{EAccept}{KbT}}}} \]

      4. lower-/.f64 41.3

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   8. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.45 \cdot 10^{-107}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 22.4% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.9 \cdot 10^{+129}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -2.9e+129)
   (* NdChar 0.5)
   (if (<= NdChar 2.8e+32) (* NaChar 0.5) (* NdChar 0.5))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -2.9e+129) {
		tmp = NdChar * 0.5;
	} else if (NdChar <= 2.8e+32) {
		tmp = NaChar * 0.5;
	} else {
		tmp = NdChar * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-2.9d+129)) then
        tmp = ndchar * 0.5d0
    else if (ndchar <= 2.8d+32) then
        tmp = nachar * 0.5d0
    else
        tmp = ndchar * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -2.9e+129) {
		tmp = NdChar * 0.5;
	} else if (NdChar <= 2.8e+32) {
		tmp = NaChar * 0.5;
	} else {
		tmp = NdChar * 0.5;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -2.9e+129:
		tmp = NdChar * 0.5
	elif NdChar <= 2.8e+32:
		tmp = NaChar * 0.5
	else:
		tmp = NdChar * 0.5
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -2.9e+129)
		tmp = Float64(NdChar * 0.5);
	elseif (NdChar <= 2.8e+32)
		tmp = Float64(NaChar * 0.5);
	else
		tmp = Float64(NdChar * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -2.9e+129)
		tmp = NdChar * 0.5;
	elseif (NdChar <= 2.8e+32)
		tmp = NaChar * 0.5;
	else
		tmp = NdChar * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -2.9e+129], N[(NdChar * 0.5), $MachinePrecision], If[LessEqual[NdChar, 2.8e+32], N[(NaChar * 0.5), $MachinePrecision], N[(NdChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.90000000000000003e129 or 2.8e32 < NdChar

   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. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. lower-+.f64 32.5

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Applied rewrites 32.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   6. Taylor expanded in NaChar around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{NdChar \cdot \frac{1}{2}} \]

      2. lower-*.f64 29.7

         \[\leadsto \color{blue}{NdChar \cdot 0.5} \]

   8. Applied rewrites 29.7%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} \]

   if -2.90000000000000003e129 < NdChar < 2.8e32

   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. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. lower-+.f64 30.3

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Applied rewrites 30.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   6. Taylor expanded in NaChar around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{NaChar \cdot \frac{1}{2}} \]

      2. lower-*.f64 25.2

         \[\leadsto \color{blue}{NaChar \cdot 0.5} \]

   8. Applied rewrites 25.2%

      \[\leadsto \color{blue}{NaChar \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 27.3% accurate, 30.7× 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. Add Preprocessing

3. Taylor expanded in KbT around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   3. lower-+.f64 31.1

      \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

5. Applied rewrites 31.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

6. Final simplification 31.1%

   \[\leadsto 0.5 \cdot \left(NdChar + NaChar\right) \]
7. Add Preprocessing

Alternative 28: 18.0% accurate, 46.0× 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. Add Preprocessing

3. Taylor expanded in KbT around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   3. lower-+.f64 31.1

      \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

5. Applied rewrites 31.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

6. Taylor expanded in NaChar around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{NaChar \cdot \frac{1}{2}} \]

   2. lower-*.f64 19.2

      \[\leadsto \color{blue}{NaChar \cdot 0.5} \]

8. Applied rewrites 19.2%

   \[\leadsto \color{blue}{NaChar \cdot 0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(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))))))