Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 18.5s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - 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(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $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. Final simplification 100.0%

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

Alternative 2: 44.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := Vef + \left(mu + \left(EDonor - Ec\right)\right)\\ t_1 := e^{\frac{Vef}{KbT}} + 1\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\frac{NaChar}{t\_1}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\frac{NdChar}{t\_1}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-245}:\\ \;\;\;\;\frac{NaChar}{e^{-\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{-279}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 10^{+118}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ Vef (+ mu (- EDonor Ec))))
        (t_1 (+ (exp (/ Vef KbT)) 1.0))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_2 -5e+113)
     (/ NaChar t_1)
     (if (<= t_2 -1e+52)
       (/ NdChar t_1)
       (if (<= t_2 -2e-245)
         (/ NaChar (+ (exp (- (/ mu KbT))) 1.0))
         (if (<= t_2 1e-279)
           (/
            NdChar
            (-
             2.0
             (/
              (fma -0.5 (/ (* t_0 t_0) KbT) (- (- (- Ec EDonor) mu) Vef))
              KbT)))
           (if (<= t_2 1e+118)
             (/ NdChar (+ (exp (/ mu 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 t_0 = Vef + (mu + (EDonor - Ec));
	double t_1 = exp((Vef / KbT)) + 1.0;
	double t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -5e+113) {
		tmp = NaChar / t_1;
	} else if (t_2 <= -1e+52) {
		tmp = NdChar / t_1;
	} else if (t_2 <= -2e-245) {
		tmp = NaChar / (exp(-(mu / KbT)) + 1.0);
	} else if (t_2 <= 1e-279) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), (((Ec - EDonor) - mu) - Vef)) / KbT));
	} else if (t_2 <= 1e+118) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Vef + Float64(mu + Float64(EDonor - Ec)))
	t_1 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -5e+113)
		tmp = Float64(NaChar / t_1);
	elseif (t_2 <= -1e+52)
		tmp = Float64(NdChar / t_1);
	elseif (t_2 <= -2e-245)
		tmp = Float64(NaChar / Float64(exp(Float64(-Float64(mu / KbT))) + 1.0));
	elseif (t_2 <= 1e-279)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Float64(Ec - EDonor) - mu) - Vef)) / KbT)));
	elseif (t_2 <= 1e+118)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+113], N[(NaChar / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -1e+52], N[(NdChar / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -2e-245], N[(NaChar / N[(N[Exp[(-N[(mu / KbT), $MachinePrecision])], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-279], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - EDonor), $MachinePrecision] - mu), $MachinePrecision] - Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+118], N[(NdChar / N[(N[Exp[N[(mu / 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 6 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))))) < -5e113

   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

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

   5. Simplified 90.7%

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

   6. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 52.6

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

   8. Simplified 52.6%

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

   if -5e113 < (+.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))))) < -9.9999999999999999e51

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 74.7

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

   5. Simplified 74.7%

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

   6. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 74.6

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

   8. Simplified 74.6%

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

   if -9.9999999999999999e51 < (+.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))))) < -1.9999999999999999e-245

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 62.4

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

   5. Simplified 62.4%

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

   6. Taylor expanded in mu around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. neg-lowering-neg.f64 50.0

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

   8. Simplified 50.0%

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

   if -1.9999999999999999e-245 < (+.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))))) < 1.00000000000000006e-279

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 98.3

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

   5. Simplified 98.3%

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

   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. --lowering--.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. /-lowering-/.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. Simplified 91.1%

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

   if 1.00000000000000006e-279 < (+.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))))) < 9.99999999999999967e117

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 67.4

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

   5. Simplified 67.4%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 52.2

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

   8. Simplified 52.2%

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

   if 9.99999999999999967e117 < (+.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 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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 67.5

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

   5. Simplified 67.5%

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

   6. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 43.1

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

   8. Simplified 43.1%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-245}:\\ \;\;\;\;\frac{NaChar}{e^{-\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-279}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right) \cdot \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{+118}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_2 := Vef + \left(mu + \left(EDonor - Ec\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-220}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{-279}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 10^{+118}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_2 (+ Vef (+ mu (- EDonor Ec)))))
   (if (<= t_1 -5e+113)
     (/ NaChar t_0)
     (if (<= t_1 -1e+52)
       (/ NdChar t_0)
       (if (<= t_1 -5e-220)
         (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
         (if (<= t_1 1e-279)
           (/
            NdChar
            (-
             2.0
             (/
              (fma -0.5 (/ (* t_2 t_2) KbT) (- (- (- Ec EDonor) mu) Vef))
              KbT)))
           (if (<= t_1 1e+118)
             (/ NdChar (+ (exp (/ mu 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 t_0 = exp((Vef / KbT)) + 1.0;
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_2 = Vef + (mu + (EDonor - Ec));
	double tmp;
	if (t_1 <= -5e+113) {
		tmp = NaChar / t_0;
	} else if (t_1 <= -1e+52) {
		tmp = NdChar / t_0;
	} else if (t_1 <= -5e-220) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else if (t_1 <= 1e-279) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_2 * t_2) / KbT), (((Ec - EDonor) - mu) - Vef)) / KbT));
	} else if (t_1 <= 1e+118) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((EAccept / 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)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_2 = Float64(Vef + Float64(mu + Float64(EDonor - Ec)))
	tmp = 0.0
	if (t_1 <= -5e+113)
		tmp = Float64(NaChar / t_0);
	elseif (t_1 <= -1e+52)
		tmp = Float64(NdChar / t_0);
	elseif (t_1 <= -5e-220)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	elseif (t_1 <= 1e-279)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Float64(Float64(Float64(Ec - EDonor) - mu) - Vef)) / KbT)));
	elseif (t_1 <= 1e+118)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+113], N[(NaChar / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -1e+52], N[(NdChar / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -5e-220], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-279], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - EDonor), $MachinePrecision] - mu), $MachinePrecision] - Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+118], N[(NdChar / N[(N[Exp[N[(mu / 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 6 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))))) < -5e113

   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

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

   5. Simplified 90.7%

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

   6. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 52.6

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

   8. Simplified 52.6%

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

   if -5e113 < (+.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))))) < -9.9999999999999999e51

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 74.7

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

   5. Simplified 74.7%

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

   6. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 74.6

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

   8. Simplified 74.6%

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

   if -9.9999999999999999e51 < (+.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))))) < -5.0000000000000002e-220

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 62.8

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

   5. Simplified 62.8%

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

   6. Taylor expanded in Ev around inf

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

      1. /-lowering-/.f64 42.3

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

   8. Simplified 42.3%

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

   if -5.0000000000000002e-220 < (+.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))))) < 1.00000000000000006e-279

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 96.6

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

   5. Simplified 96.6%

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

   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. --lowering--.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. /-lowering-/.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. Simplified 88.0%

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

   if 1.00000000000000006e-279 < (+.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))))) < 9.99999999999999967e117

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 67.4

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

   5. Simplified 67.4%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 52.2

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

   8. Simplified 52.2%

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

   if 9.99999999999999967e117 < (+.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 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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 67.5

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

   5. Simplified 67.5%

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

   6. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 43.1

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

   8. Simplified 43.1%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-220}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-279}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right) \cdot \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{+118}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{-1 - e^{\frac{Vef}{KbT}}} \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_2 := Vef + \left(mu + \left(EDonor - Ec\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-245}:\\ \;\;\;\;\frac{NaChar}{e^{-\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{-279}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (/ -1.0 (- -1.0 (exp (/ Vef KbT)))) (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_2 (+ Vef (+ mu (- EDonor Ec)))))
   (if (<= t_1 -5e+22)
     t_0
     (if (<= t_1 -2e-245)
       (/ NaChar (+ (exp (- (/ mu KbT))) 1.0))
       (if (<= t_1 1e-279)
         (/
          NdChar
          (-
           2.0
           (/
            (fma -0.5 (/ (* t_2 t_2) KbT) (- (- (- Ec EDonor) mu) Vef))
            KbT)))
         (if (<= t_1 2e+112) (/ NdChar (+ (exp (/ 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 = (-1.0 / (-1.0 - exp((Vef / KbT)))) * (NdChar + NaChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_2 = Vef + (mu + (EDonor - Ec));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = t_0;
	} else if (t_1 <= -2e-245) {
		tmp = NaChar / (exp(-(mu / KbT)) + 1.0);
	} else if (t_1 <= 1e-279) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_2 * t_2) / KbT), (((Ec - EDonor) - mu) - Vef)) / KbT));
	} else if (t_1 <= 2e+112) {
		tmp = NdChar / (exp((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(-1.0 / Float64(-1.0 - exp(Float64(Vef / KbT)))) * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_2 = Float64(Vef + Float64(mu + Float64(EDonor - Ec)))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = t_0;
	elseif (t_1 <= -2e-245)
		tmp = Float64(NaChar / Float64(exp(Float64(-Float64(mu / KbT))) + 1.0));
	elseif (t_1 <= 1e-279)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Float64(Float64(Float64(Ec - EDonor) - mu) - Vef)) / KbT)));
	elseif (t_1 <= 2e+112)
		tmp = Float64(NdChar / Float64(exp(Float64(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[(N[(-1.0 / N[(-1.0 - N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+22], t$95$0, If[LessEqual[t$95$1, -2e-245], N[(NaChar / N[(N[Exp[(-N[(mu / KbT), $MachinePrecision])], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-279], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - EDonor), $MachinePrecision] - mu), $MachinePrecision] - Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+112], N[(NdChar / N[(N[Exp[N[(mu / 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))))) < -4.9999999999999996e22 or 1.9999999999999999e112 < (+.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 Vef around inf

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

   5. Simplified 84.2%

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

   6. Taylor expanded in Vef around inf

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

   8. Simplified 69.0%

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

      1. div-inv N/A

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

      2. div-inv N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 69.0

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

   10. Applied egg-rr 69.0%

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

   if -4.9999999999999996e22 < (+.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))))) < -1.9999999999999999e-245

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 62.0

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

   5. Simplified 62.0%

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

   6. Taylor expanded in mu around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. neg-lowering-neg.f64 48.8

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

   8. Simplified 48.8%

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

   if -1.9999999999999999e-245 < (+.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))))) < 1.00000000000000006e-279

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 98.3

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

   5. Simplified 98.3%

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

   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. --lowering--.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. /-lowering-/.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. Simplified 91.1%

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

   if 1.00000000000000006e-279 < (+.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))))) < 1.9999999999999999e112

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 53.6

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

   8. Simplified 53.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{-1 - e^{\frac{Vef}{KbT}}} \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-245}:\\ \;\;\;\;\frac{NaChar}{e^{-\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-279}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right) \cdot \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - e^{\frac{Vef}{KbT}}} \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ t_1 := t\_0 + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ t_2 := t\_0 + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-220}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+53}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))
        (t_1 (+ t_0 (/ NaChar (+ (exp (/ Vef KbT)) 1.0))))
        (t_2
         (+
          t_0
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_2 -0.05)
     t_1
     (if (<= t_2 -5e-220)
       (/ NaChar (+ (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)) 1.0))
       (if (<= t_2 4e+53)
         (/ NdChar (+ (exp (/ (+ (+ Vef EDonor) (- mu Ec)) KbT)) 1.0))
         t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	double t_1 = t_0 + (NaChar / (exp((Vef / KbT)) + 1.0));
	double t_2 = t_0 + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = t_1;
	} else if (t_2 <= -5e-220) {
		tmp = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	} else if (t_2 <= 4e+53) {
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)
    t_1 = t_0 + (nachar / (exp((vef / kbt)) + 1.0d0))
    t_2 = t_0 + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_2 <= (-0.05d0)) then
        tmp = t_1
    else if (t_2 <= (-5d-220)) then
        tmp = nachar / (exp((((ev + eaccept) + (vef - mu)) / kbt)) + 1.0d0)
    else if (t_2 <= 4d+53) then
        tmp = ndchar / (exp((((vef + edonor) + (mu - ec)) / kbt)) + 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	double t_1 = t_0 + (NaChar / (Math.exp((Vef / KbT)) + 1.0));
	double t_2 = t_0 + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = t_1;
	} else if (t_2 <= -5e-220) {
		tmp = NaChar / (Math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	} else if (t_2 <= 4e+53) {
		tmp = NdChar / (Math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)
	t_1 = t_0 + (NaChar / (math.exp((Vef / KbT)) + 1.0))
	t_2 = t_0 + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_2 <= -0.05:
		tmp = t_1
	elif t_2 <= -5e-220:
		tmp = NaChar / (math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0)
	elif t_2 <= 4e+53:
		tmp = NdChar / (math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0))
	t_1 = Float64(t_0 + Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)))
	t_2 = Float64(t_0 + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = t_1;
	elseif (t_2 <= -5e-220)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)) + 1.0));
	elseif (t_2 <= 4e+53)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Vef + EDonor) + Float64(mu - Ec)) / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	t_1 = t_0 + (NaChar / (exp((Vef / KbT)) + 1.0));
	t_2 = t_0 + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_2 <= -0.05)
		tmp = t_1;
	elseif (t_2 <= -5e-220)
		tmp = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	elseif (t_2 <= 4e+53)
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.05], t$95$1, If[LessEqual[t$95$2, -5e-220], N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+53], N[(NdChar / N[(N[Exp[N[(N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision] / 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))))) < -0.050000000000000003 or 4e53 < (+.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 Vef around inf

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

   5. Simplified 82.7%

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

   if -0.050000000000000003 < (+.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))))) < -5.0000000000000002e-220

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 64.5

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

   5. Simplified 64.5%

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

   if -5.0000000000000002e-220 < (+.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))))) < 4e53

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 84.6

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

   5. Simplified 84.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -0.05:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-220}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{+53}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_1 := \frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{-1 - e^{\frac{Vef}{KbT}}} \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+118}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_1 (/ NaChar (+ (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)) 1.0))))
   (if (<= t_0 -5e+22)
     (* (/ -1.0 (- -1.0 (exp (/ Vef KbT)))) (+ NdChar NaChar))
     (if (<= t_0 1e-35)
       t_1
       (if (<= t_0 1e+118) (/ NdChar (+ (exp (/ mu KbT)) 1.0)) t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_1 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (t_0 <= -5e+22) {
		tmp = (-1.0 / (-1.0 - exp((Vef / KbT)))) * (NdChar + NaChar);
	} else if (t_0 <= 1e-35) {
		tmp = t_1;
	} else if (t_0 <= 1e+118) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    t_1 = nachar / (exp((((ev + eaccept) + (vef - mu)) / kbt)) + 1.0d0)
    if (t_0 <= (-5d+22)) then
        tmp = ((-1.0d0) / ((-1.0d0) - exp((vef / kbt)))) * (ndchar + nachar)
    else if (t_0 <= 1d-35) then
        tmp = t_1
    else if (t_0 <= 1d+118) then
        tmp = ndchar / (exp((mu / kbt)) + 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_1 = NaChar / (Math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (t_0 <= -5e+22) {
		tmp = (-1.0 / (-1.0 - Math.exp((Vef / KbT)))) * (NdChar + NaChar);
	} else if (t_0 <= 1e-35) {
		tmp = t_1;
	} else if (t_0 <= 1e+118) {
		tmp = NdChar / (Math.exp((mu / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	t_1 = NaChar / (math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0)
	tmp = 0
	if t_0 <= -5e+22:
		tmp = (-1.0 / (-1.0 - math.exp((Vef / KbT)))) * (NdChar + NaChar)
	elif t_0 <= 1e-35:
		tmp = t_1
	elif t_0 <= 1e+118:
		tmp = NdChar / (math.exp((mu / KbT)) + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_1 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)) + 1.0))
	tmp = 0.0
	if (t_0 <= -5e+22)
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - exp(Float64(Vef / KbT)))) * Float64(NdChar + NaChar));
	elseif (t_0 <= 1e-35)
		tmp = t_1;
	elseif (t_0 <= 1e+118)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	t_1 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	tmp = 0.0;
	if (t_0 <= -5e+22)
		tmp = (-1.0 / (-1.0 - exp((Vef / KbT)))) * (NdChar + NaChar);
	elseif (t_0 <= 1e-35)
		tmp = t_1;
	elseif (t_0 <= 1e+118)
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+22], N[(N[(-1.0 / N[(-1.0 - N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-35], t$95$1, If[LessEqual[t$95$0, 1e+118], N[(NdChar / N[(N[Exp[N[(mu / 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))))) < -4.9999999999999996e22

   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

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

   5. Simplified 88.6%

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

   6. Taylor expanded in Vef around inf

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

   8. Simplified 73.8%

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

      1. div-inv N/A

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

      2. div-inv N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 73.8

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

   10. Applied egg-rr 73.8%

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

   if -4.9999999999999996e22 < (+.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))))) < 1.00000000000000001e-35 or 9.99999999999999967e117 < (+.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 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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 73.7

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

   5. Simplified 73.7%

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

   if 1.00000000000000001e-35 < (+.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))))) < 9.99999999999999967e117

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 77.4

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

   5. Simplified 77.4%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 57.4

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

   8. Simplified 57.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{-1 - e^{\frac{Vef}{KbT}}} \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-35}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{+118}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := Vef + \left(mu + \left(EDonor - Ec\right)\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_2 := \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-245}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ Vef (+ mu (- EDonor Ec))))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_2 (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))))
   (if (<= t_1 -5e+22)
     (* (+ NdChar NaChar) 0.5)
     (if (<= t_1 -2e-245)
       t_2
       (if (<= t_1 0.0)
         (/
          NdChar
          (-
           2.0
           (/
            (fma -0.5 (/ (* t_0 t_0) KbT) (- (- (- Ec EDonor) mu) Vef))
            KbT)))
         t_2)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Vef + (mu + (EDonor - Ec));
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_2 = NaChar / (exp((EAccept / KbT)) + 1.0);
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = (NdChar + NaChar) * 0.5;
	} else if (t_1 <= -2e-245) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), (((Ec - EDonor) - mu) - Vef)) / KbT));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Vef + Float64(mu + Float64(EDonor - Ec)))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_2 = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = Float64(Float64(NdChar + NaChar) * 0.5);
	elseif (t_1 <= -2e-245)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Float64(Ec - EDonor) - mu) - Vef)) / KbT)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+22], N[(N[(NdChar + NaChar), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, -2e-245], t$95$2, If[LessEqual[t$95$1, 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 - EDonor), $MachinePrecision] - mu), $MachinePrecision] - Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

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))))) < -4.9999999999999996e22

   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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 45.3

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

   5. Simplified 45.3%

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

   if -4.9999999999999996e22 < (+.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))))) < -1.9999999999999999e-245 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 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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 54.1

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

   5. Simplified 54.1%

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

   6. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 36.1

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

   8. Simplified 36.1%

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

   if -1.9999999999999999e-245 < (+.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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 100.0

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

   5. Simplified 100.0%

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

   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. --lowering--.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. /-lowering-/.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. Simplified 92.7%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-245}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right) \cdot \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := Vef + \left(mu + \left(EDonor - Ec\right)\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-220}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ Vef (+ mu (- EDonor Ec))))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -5e-220)
     (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
     (if (<= t_1 0.0)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- (- Ec EDonor) mu) Vef)) KbT)))
       (/ 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 = Vef + (mu + (EDonor - Ec));
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-220) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), (((Ec - EDonor) - mu) - Vef)) / KbT));
	} else {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Vef + Float64(mu + Float64(EDonor - Ec)))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e-220)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Float64(Ec - EDonor) - mu) - Vef)) / KbT)));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-220], N[(NaChar / N[(N[Exp[N[(Ev / 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$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - EDonor), $MachinePrecision] - mu), $MachinePrecision] - Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $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 (+.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))))) < -5.0000000000000002e-220

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 57.7

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

   5. Simplified 57.7%

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

   6. Taylor expanded in Ev around inf

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

      1. /-lowering-/.f64 42.6

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

   8. Simplified 42.6%

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

   if -5.0000000000000002e-220 < (+.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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 98.3

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

   5. Simplified 98.3%

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

   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. --lowering--.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. /-lowering-/.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. Simplified 89.5%

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

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

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 50.6

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

   5. Simplified 50.6%

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

   6. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 33.5

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

   8. Simplified 33.5%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-220}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right) \cdot \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := Vef + \left(mu + \left(EDonor - Ec\right)\right)\\ t_1 := \left(NdChar + NaChar\right) \cdot 0.5\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-208}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-143}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ Vef (+ mu (- EDonor Ec))))
        (t_1 (* (+ NdChar NaChar) 0.5))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_2 -2e-208)
     t_1
     (if (<= t_2 1e-143)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- (- Ec EDonor) mu) Vef)) KbT)))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Vef + (mu + (EDonor - Ec));
	double t_1 = (NdChar + NaChar) * 0.5;
	double t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -2e-208) {
		tmp = t_1;
	} else if (t_2 <= 1e-143) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), (((Ec - EDonor) - mu) - Vef)) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Vef + Float64(mu + Float64(EDonor - Ec)))
	t_1 = Float64(Float64(NdChar + NaChar) * 0.5)
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -2e-208)
		tmp = t_1;
	elseif (t_2 <= 1e-143)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Float64(Ec - EDonor) - mu) - Vef)) / 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[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar + NaChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-208], t$95$1, If[LessEqual[t$95$2, 1e-143], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(Ec - EDonor), $MachinePrecision] - mu), $MachinePrecision] - Vef), $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))))) < -2.0000000000000002e-208 or 9.9999999999999995e-144 < (+.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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 35.0

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

   5. Simplified 35.0%

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

   if -2.0000000000000002e-208 < (+.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))))) < 9.9999999999999995e-144

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 91.5

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

   5. Simplified 91.5%

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

   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. --lowering--.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. /-lowering-/.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. Simplified 75.8%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-143}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right) \cdot \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)}{KbT}, \left(\left(Ec - EDonor\right) - mu\right) - Vef\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NdChar + NaChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-279}:\\ \;\;\;\;\frac{0.125 \cdot \left(NaChar \cdot \left(NaChar \cdot NaChar\right)\right)}{\mathsf{fma}\left(NaChar \cdot NaChar, 0.25, \left(NdChar \cdot 0.5\right) \cdot \left(NdChar \cdot 0.5\right) - \left(NdChar \cdot 0.5\right) \cdot \left(NaChar \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NdChar NaChar) 0.5))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-208)
     t_0
     (if (<= t_1 1e-279)
       (/
        (* 0.125 (* NaChar (* NaChar NaChar)))
        (fma
         (* NaChar NaChar)
         0.25
         (-
          (* (* NdChar 0.5) (* NdChar 0.5))
          (* (* NdChar 0.5) (* NaChar 0.5)))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-208) {
		tmp = t_0;
	} else if (t_1 <= 1e-279) {
		tmp = (0.125 * (NaChar * (NaChar * NaChar))) / fma((NaChar * NaChar), 0.25, (((NdChar * 0.5) * (NdChar * 0.5)) - ((NdChar * 0.5) * (NaChar * 0.5))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar + NaChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-208)
		tmp = t_0;
	elseif (t_1 <= 1e-279)
		tmp = Float64(Float64(0.125 * Float64(NaChar * Float64(NaChar * NaChar))) / fma(Float64(NaChar * NaChar), 0.25, Float64(Float64(Float64(NdChar * 0.5) * Float64(NdChar * 0.5)) - Float64(Float64(NdChar * 0.5) * Float64(NaChar * 0.5)))));
	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[(NdChar + NaChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-208], t$95$0, If[LessEqual[t$95$1, 1e-279], N[(N[(0.125 * N[(NaChar * N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(NaChar * NaChar), $MachinePrecision] * 0.25 + N[(N[(N[(NdChar * 0.5), $MachinePrecision] * N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(NdChar * 0.5), $MachinePrecision] * N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]), $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))))) < -2.0000000000000002e-208 or 1.00000000000000006e-279 < (+.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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 33.5

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

   5. Simplified 33.5%

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

   if -2.0000000000000002e-208 < (+.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))))) < 1.00000000000000006e-279

   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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 2.5

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

   5. Simplified 2.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(NaChar \cdot \frac{1}{2}\right)}^{3} + {\left(NdChar \cdot \frac{1}{2}\right)}^{3}}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)}} \]

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

         \[\leadsto \color{blue}{\frac{{\left(NaChar \cdot \frac{1}{2}\right)}^{3} + {\left(NdChar \cdot \frac{1}{2}\right)}^{3}}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)}} \]

      4. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{NaChar}^{3} \cdot {\frac{1}{2}}^{3}} + {\left(NdChar \cdot \frac{1}{2}\right)}^{3}}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({NaChar}^{3}, {\frac{1}{2}}^{3}, {\left(NdChar \cdot \frac{1}{2}\right)}^{3}\right)}}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      6. cube-mult N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{NaChar \cdot \left(NaChar \cdot NaChar\right)}, {\frac{1}{2}}^{3}, {\left(NdChar \cdot \frac{1}{2}\right)}^{3}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{NaChar \cdot \left(NaChar \cdot NaChar\right)}, {\frac{1}{2}}^{3}, {\left(NdChar \cdot \frac{1}{2}\right)}^{3}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \color{blue}{\left(NaChar \cdot NaChar\right)}, {\frac{1}{2}}^{3}, {\left(NdChar \cdot \frac{1}{2}\right)}^{3}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \color{blue}{\frac{1}{8}}, {\left(NdChar \cdot \frac{1}{2}\right)}^{3}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      10. unpow-prod-down N/A

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \color{blue}{{NdChar}^{3} \cdot {\frac{1}{2}}^{3}}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \color{blue}{{NdChar}^{3} \cdot {\frac{1}{2}}^{3}}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      12. cube-mult N/A

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \color{blue}{\left(NdChar \cdot \left(NdChar \cdot NdChar\right)\right)} \cdot {\frac{1}{2}}^{3}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \color{blue}{\left(NdChar \cdot \left(NdChar \cdot NdChar\right)\right)} \cdot {\frac{1}{2}}^{3}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \left(NdChar \cdot \color{blue}{\left(NdChar \cdot NdChar\right)}\right) \cdot {\frac{1}{2}}^{3}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \left(NdChar \cdot \left(NdChar \cdot NdChar\right)\right) \cdot \color{blue}{\frac{1}{8}}\right)}{\left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NaChar \cdot \frac{1}{2}\right) + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      16. swap-sqr N/A

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \left(NdChar \cdot \left(NdChar \cdot NdChar\right)\right) \cdot \frac{1}{8}\right)}{\color{blue}{\left(NaChar \cdot NaChar\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} + \left(\left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \left(NdChar \cdot \left(NdChar \cdot NdChar\right)\right) \cdot \frac{1}{8}\right)}{\color{blue}{\mathsf{fma}\left(NaChar \cdot NaChar, \frac{1}{2} \cdot \frac{1}{2}, \left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)}} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \left(NdChar \cdot \left(NdChar \cdot NdChar\right)\right) \cdot \frac{1}{8}\right)}{\mathsf{fma}\left(\color{blue}{NaChar \cdot NaChar}, \frac{1}{2} \cdot \frac{1}{2}, \left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), \frac{1}{8}, \left(NdChar \cdot \left(NdChar \cdot NdChar\right)\right) \cdot \frac{1}{8}\right)}{\mathsf{fma}\left(NaChar \cdot NaChar, \color{blue}{\frac{1}{4}}, \left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

   7. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(NaChar \cdot \left(NaChar \cdot NaChar\right), 0.125, \left(NdChar \cdot \left(NdChar \cdot NdChar\right)\right) \cdot 0.125\right)}{\mathsf{fma}\left(NaChar \cdot NaChar, 0.25, \left(NdChar \cdot 0.5\right) \cdot \left(NdChar \cdot 0.5\right) - \left(NaChar \cdot 0.5\right) \cdot \left(NdChar \cdot 0.5\right)\right)}} \]

   8. Taylor expanded in NaChar around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot {NaChar}^{3}}}{\mathsf{fma}\left(NaChar \cdot NaChar, \frac{1}{4}, \left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot {NaChar}^{3}}}{\mathsf{fma}\left(NaChar \cdot NaChar, \frac{1}{4}, \left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      2. cube-mult N/A

         \[\leadsto \frac{\frac{1}{8} \cdot \color{blue}{\left(NaChar \cdot \left(NaChar \cdot NaChar\right)\right)}}{\mathsf{fma}\left(NaChar \cdot NaChar, \frac{1}{4}, \left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot \left(NaChar \cdot \color{blue}{{NaChar}^{2}}\right)}{\mathsf{fma}\left(NaChar \cdot NaChar, \frac{1}{4}, \left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

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

         \[\leadsto \frac{\frac{1}{8} \cdot \color{blue}{\left(NaChar \cdot {NaChar}^{2}\right)}}{\mathsf{fma}\left(NaChar \cdot NaChar, \frac{1}{4}, \left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot \left(NaChar \cdot \color{blue}{\left(NaChar \cdot NaChar\right)}\right)}{\mathsf{fma}\left(NaChar \cdot NaChar, \frac{1}{4}, \left(NdChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right) - \left(NaChar \cdot \frac{1}{2}\right) \cdot \left(NdChar \cdot \frac{1}{2}\right)\right)} \]

      6. *-lowering-*.f64 43.8

         \[\leadsto \frac{0.125 \cdot \left(NaChar \cdot \color{blue}{\left(NaChar \cdot NaChar\right)}\right)}{\mathsf{fma}\left(NaChar \cdot NaChar, 0.25, \left(NdChar \cdot 0.5\right) \cdot \left(NdChar \cdot 0.5\right) - \left(NaChar \cdot 0.5\right) \cdot \left(NdChar \cdot 0.5\right)\right)} \]

   10. Simplified 43.8%

       \[\leadsto \frac{\color{blue}{0.125 \cdot \left(NaChar \cdot \left(NaChar \cdot NaChar\right)\right)}}{\mathsf{fma}\left(NaChar \cdot NaChar, 0.25, \left(NdChar \cdot 0.5\right) \cdot \left(NdChar \cdot 0.5\right) - \left(NaChar \cdot 0.5\right) \cdot \left(NdChar \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-279}:\\ \;\;\;\;\frac{0.125 \cdot \left(NaChar \cdot \left(NaChar \cdot NaChar\right)\right)}{\mathsf{fma}\left(NaChar \cdot NaChar, 0.25, \left(NdChar \cdot 0.5\right) \cdot \left(NdChar \cdot 0.5\right) - \left(NdChar \cdot 0.5\right) \cdot \left(NaChar \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NdChar + NaChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\frac{NaChar}{\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \left(\frac{Vef}{KbT} - \frac{mu}{KbT}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NdChar NaChar) 0.5))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -5e-92)
     t_0
     (if (<= t_1 2e-230)
       (/
        NaChar
        (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Ev KbT) (- (/ Vef KbT) (/ mu KbT)))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-92) {
		tmp = t_0;
	} else if (t_1 <= 2e-230) {
		tmp = NaChar / ((2.0 + (EAccept / KbT)) + ((Ev / KbT) + ((Vef / KbT) - (mu / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (ndchar + nachar) * 0.5d0
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-5d-92)) then
        tmp = t_0
    else if (t_1 <= 2d-230) then
        tmp = nachar / ((2.0d0 + (eaccept / kbt)) + ((ev / kbt) + ((vef / kbt) - (mu / kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-92) {
		tmp = t_0;
	} else if (t_1 <= 2e-230) {
		tmp = NaChar / ((2.0 + (EAccept / KbT)) + ((Ev / KbT) + ((Vef / KbT) - (mu / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar + NaChar) * 0.5
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -5e-92:
		tmp = t_0
	elif t_1 <= 2e-230:
		tmp = NaChar / ((2.0 + (EAccept / KbT)) + ((Ev / KbT) + ((Vef / KbT) - (mu / KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar + NaChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e-92)
		tmp = t_0;
	elseif (t_1 <= 2e-230)
		tmp = Float64(NaChar / Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Ev / KbT) + Float64(Float64(Vef / KbT) - Float64(mu / KbT)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar + NaChar) * 0.5;
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -5e-92)
		tmp = t_0;
	elseif (t_1 <= 2e-230)
		tmp = NaChar / ((2.0 + (EAccept / KbT)) + ((Ev / KbT) + ((Vef / KbT) - (mu / KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar + NaChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-92], t$95$0, If[LessEqual[t$95$1, 2e-230], N[(NaChar / N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $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))))) < -5.00000000000000011e-92 or 2.00000000000000009e-230 < (+.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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 34.6

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

   5. Simplified 34.6%

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

   if -5.00000000000000011e-92 < (+.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))))) < 2.00000000000000009e-230

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 85.6

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

   5. Simplified 85.6%

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

   6. Taylor expanded in KbT around inf

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

      1. associate--l+ N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

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

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      12. /-lowering-/.f64 38.3

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

   8. Simplified 38.3%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-92}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\frac{NaChar}{\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \left(\frac{Vef}{KbT} - \frac{mu}{KbT}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NdChar + NaChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-259}:\\ \;\;\;\;Vef \cdot \left(Ec \cdot \left(-0.25 \cdot \frac{NdChar}{Vef \cdot \left(-KbT\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NdChar NaChar) 0.5))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -5e-270)
     t_0
     (if (<= t_1 5e-259)
       (* Vef (* Ec (* -0.25 (/ NdChar (* Vef (- KbT))))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-270) {
		tmp = t_0;
	} else if (t_1 <= 5e-259) {
		tmp = Vef * (Ec * (-0.25 * (NdChar / (Vef * -KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (ndchar + nachar) * 0.5d0
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-5d-270)) then
        tmp = t_0
    else if (t_1 <= 5d-259) then
        tmp = vef * (ec * ((-0.25d0) * (ndchar / (vef * -kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-270) {
		tmp = t_0;
	} else if (t_1 <= 5e-259) {
		tmp = Vef * (Ec * (-0.25 * (NdChar / (Vef * -KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar + NaChar) * 0.5
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -5e-270:
		tmp = t_0
	elif t_1 <= 5e-259:
		tmp = Vef * (Ec * (-0.25 * (NdChar / (Vef * -KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar + NaChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e-270)
		tmp = t_0;
	elseif (t_1 <= 5e-259)
		tmp = Float64(Vef * Float64(Ec * Float64(-0.25 * Float64(NdChar / Float64(Vef * Float64(-KbT))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar + NaChar) * 0.5;
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -5e-270)
		tmp = t_0;
	elseif (t_1 <= 5e-259)
		tmp = Vef * (Ec * (-0.25 * (NdChar / (Vef * -KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar + NaChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-270], t$95$0, If[LessEqual[t$95$1, 5e-259], N[(Vef * N[(Ec * N[(-0.25 * N[(NdChar / N[(Vef * (-KbT)), $MachinePrecision]), $MachinePrecision]), $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))))) < -4.9999999999999998e-270 or 4.99999999999999977e-259 < (+.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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 33.0

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

   5. Simplified 33.0%

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

   if -4.9999999999999998e-270 < (+.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))))) < 4.99999999999999977e-259

   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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 4.2

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

   5. Simplified 4.2%

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

   6. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

   8. Simplified 1.8%

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

   9. Taylor expanded in Vef around -inf

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

       1. mul-1-neg N/A

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

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

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

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

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

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

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

       8. associate-*r/ N/A

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

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

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

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

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

   11. Simplified 1.7%

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

   12. Taylor expanded in Ec around inf

       \[\leadsto \mathsf{neg}\left(Vef \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{Ec \cdot NdChar}{KbT \cdot Vef}\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(Vef \cdot \color{blue}{\left(\frac{Ec \cdot NdChar}{KbT \cdot Vef} \cdot \frac{-1}{4}\right)}\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{neg}\left(Vef \cdot \left(\color{blue}{\left(Ec \cdot \frac{NdChar}{KbT \cdot Vef}\right)} \cdot \frac{-1}{4}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(Vef \cdot \color{blue}{\left(Ec \cdot \left(\frac{NdChar}{KbT \cdot Vef} \cdot \frac{-1}{4}\right)\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(Vef \cdot \left(Ec \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar}{KbT \cdot Vef}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{neg}\left(Vef \cdot \color{blue}{\left(Ec \cdot \left(\frac{-1}{4} \cdot \frac{NdChar}{KbT \cdot Vef}\right)\right)}\right) \]

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

          \[\leadsto \mathsf{neg}\left(Vef \cdot \left(Ec \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar}{KbT \cdot Vef}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{neg}\left(Vef \cdot \left(Ec \cdot \left(\frac{-1}{4} \cdot \color{blue}{\frac{NdChar}{KbT \cdot Vef}}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(Vef \cdot \left(Ec \cdot \left(\frac{-1}{4} \cdot \frac{NdChar}{\color{blue}{Vef \cdot KbT}}\right)\right)\right) \]

       9. *-lowering-*.f64 28.3

          \[\leadsto -Vef \cdot \left(Ec \cdot \left(-0.25 \cdot \frac{NdChar}{\color{blue}{Vef \cdot KbT}}\right)\right) \]

   14. Simplified 28.3%

       \[\leadsto -Vef \cdot \color{blue}{\left(Ec \cdot \left(-0.25 \cdot \frac{NdChar}{Vef \cdot KbT}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-259}:\\ \;\;\;\;Vef \cdot \left(Ec \cdot \left(-0.25 \cdot \frac{NdChar}{Vef \cdot \left(-KbT\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NdChar + NaChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-157}:\\ \;\;\;\;Vef \cdot \frac{NaChar \cdot \left(--0.5\right)}{Vef}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NdChar NaChar) 0.5))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -5e+22)
     t_0
     (if (<= t_1 4e-157) (* Vef (/ (* NaChar (- -0.5)) 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 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = t_0;
	} else if (t_1 <= 4e-157) {
		tmp = Vef * ((NaChar * -(-0.5)) / 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 = (ndchar + nachar) * 0.5d0
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-5d+22)) then
        tmp = t_0
    else if (t_1 <= 4d-157) then
        tmp = vef * ((nachar * -(-0.5d0)) / 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 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = t_0;
	} else if (t_1 <= 4e-157) {
		tmp = Vef * ((NaChar * -(-0.5)) / Vef);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar + NaChar) * 0.5
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -5e+22:
		tmp = t_0
	elif t_1 <= 4e-157:
		tmp = Vef * ((NaChar * -(-0.5)) / Vef)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar + NaChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = t_0;
	elseif (t_1 <= 4e-157)
		tmp = Float64(Vef * Float64(Float64(NaChar * Float64(-(-0.5))) / 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 = (NdChar + NaChar) * 0.5;
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -5e+22)
		tmp = t_0;
	elseif (t_1 <= 4e-157)
		tmp = Vef * ((NaChar * -(-0.5)) / 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[(N[(NdChar + NaChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+22], t$95$0, If[LessEqual[t$95$1, 4e-157], N[(Vef * N[(N[(NaChar * (--0.5)), $MachinePrecision] / 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))))) < -4.9999999999999996e22 or 3.99999999999999977e-157 < (+.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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 37.6

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

   5. Simplified 37.6%

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

   if -4.9999999999999996e22 < (+.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))))) < 3.99999999999999977e-157

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 21.3

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

   5. Simplified 21.3%

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

   6. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

   8. Simplified 7.8%

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

   9. Taylor expanded in Vef around -inf

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

       1. mul-1-neg N/A

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

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

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

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

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

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

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

       8. associate-*r/ N/A

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

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

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

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

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

   11. Simplified 6.7%

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

   12. Taylor expanded in NdChar around 0

       \[\leadsto \mathsf{neg}\left(Vef \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{NaChar}{Vef}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{neg}\left(Vef \cdot \color{blue}{\frac{\frac{-1}{2} \cdot NaChar}{Vef}}\right) \]

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

          \[\leadsto \mathsf{neg}\left(Vef \cdot \color{blue}{\frac{\frac{-1}{2} \cdot NaChar}{Vef}}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(Vef \cdot \frac{\color{blue}{NaChar \cdot \frac{-1}{2}}}{Vef}\right) \]

       4. *-lowering-*.f64 23.4

          \[\leadsto -Vef \cdot \frac{\color{blue}{NaChar \cdot -0.5}}{Vef} \]

   14. Simplified 23.4%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{-157}:\\ \;\;\;\;Vef \cdot \frac{NaChar \cdot \left(--0.5\right)}{Vef}\\ \mathbf{else}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NdChar + NaChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-279}:\\ \;\;\;\;\frac{-0.25 \cdot \left(NaChar \cdot EAccept\right)}{KbT}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NdChar NaChar) 0.5))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-208)
     t_0
     (if (<= t_1 1e-279) (/ (* -0.25 (* NaChar EAccept)) KbT) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-208) {
		tmp = t_0;
	} else if (t_1 <= 1e-279) {
		tmp = (-0.25 * (NaChar * EAccept)) / KbT;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (ndchar + nachar) * 0.5d0
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-2d-208)) then
        tmp = t_0
    else if (t_1 <= 1d-279) then
        tmp = ((-0.25d0) * (nachar * eaccept)) / kbt
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-208) {
		tmp = t_0;
	} else if (t_1 <= 1e-279) {
		tmp = (-0.25 * (NaChar * EAccept)) / KbT;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar + NaChar) * 0.5
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -2e-208:
		tmp = t_0
	elif t_1 <= 1e-279:
		tmp = (-0.25 * (NaChar * EAccept)) / KbT
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar + NaChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-208)
		tmp = t_0;
	elseif (t_1 <= 1e-279)
		tmp = Float64(Float64(-0.25 * Float64(NaChar * EAccept)) / KbT);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar + NaChar) * 0.5;
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-208)
		tmp = t_0;
	elseif (t_1 <= 1e-279)
		tmp = (-0.25 * (NaChar * EAccept)) / KbT;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar + NaChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-208], t$95$0, If[LessEqual[t$95$1, 1e-279], N[(N[(-0.25 * N[(NaChar * EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $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))))) < -2.0000000000000002e-208 or 1.00000000000000006e-279 < (+.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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 33.5

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

   5. Simplified 33.5%

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

   if -2.0000000000000002e-208 < (+.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))))) < 1.00000000000000006e-279

   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

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

   5. Simplified 1.5%

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

   6. Taylor expanded in EAccept around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{EAccept \cdot NaChar}{KbT}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{4} \cdot \left(EAccept \cdot NaChar\right)}{KbT}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{-1}{4} \cdot \left(EAccept \cdot NaChar\right)}{KbT}} \]

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

         \[\leadsto \frac{\color{blue}{\frac{-1}{4} \cdot \left(EAccept \cdot NaChar\right)}}{KbT} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \color{blue}{\left(NaChar \cdot EAccept\right)}}{KbT} \]

      5. *-lowering-*.f64 22.2

         \[\leadsto \frac{-0.25 \cdot \color{blue}{\left(NaChar \cdot EAccept\right)}}{KbT} \]

   8. Simplified 22.2%

      \[\leadsto \color{blue}{\frac{-0.25 \cdot \left(NaChar \cdot EAccept\right)}{KbT}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-279}:\\ \;\;\;\;\frac{-0.25 \cdot \left(NaChar \cdot EAccept\right)}{KbT}\\ \mathbf{else}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ t_1 := e^{\frac{Vef}{KbT}} + 1\\ t_2 := \frac{NdChar}{t\_1}\\ t_3 := \frac{NaChar}{t\_1}\\ \mathbf{if}\;Vef \leq -1.95 \cdot 10^{+188}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;Vef \leq -6.5 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Vef \leq -2.9 \cdot 10^{-260}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 9.8 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 5.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 5.8 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 1.2 \cdot 10^{+250}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)))
        (t_1 (+ (exp (/ Vef KbT)) 1.0))
        (t_2 (/ NdChar t_1))
        (t_3 (/ NaChar t_1)))
   (if (<= Vef -1.95e+188)
     t_3
     (if (<= Vef -6.5e+138)
       t_2
       (if (<= Vef -2.9e-260)
         (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
         (if (<= Vef 9.8e-286)
           t_0
           (if (<= Vef 5.6e-137)
             (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
             (if (<= Vef 5.8e+47) t_0 (if (<= Vef 1.2e+250) t_3 t_2)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double t_1 = exp((Vef / KbT)) + 1.0;
	double t_2 = NdChar / t_1;
	double t_3 = NaChar / t_1;
	double tmp;
	if (Vef <= -1.95e+188) {
		tmp = t_3;
	} else if (Vef <= -6.5e+138) {
		tmp = t_2;
	} else if (Vef <= -2.9e-260) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else if (Vef <= 9.8e-286) {
		tmp = t_0;
	} else if (Vef <= 5.6e-137) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (Vef <= 5.8e+47) {
		tmp = t_0;
	} else if (Vef <= 1.2e+250) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ndchar / (exp((edonor / kbt)) + 1.0d0)
    t_1 = exp((vef / kbt)) + 1.0d0
    t_2 = ndchar / t_1
    t_3 = nachar / t_1
    if (vef <= (-1.95d+188)) then
        tmp = t_3
    else if (vef <= (-6.5d+138)) then
        tmp = t_2
    else if (vef <= (-2.9d-260)) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else if (vef <= 9.8d-286) then
        tmp = t_0
    else if (vef <= 5.6d-137) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else if (vef <= 5.8d+47) then
        tmp = t_0
    else if (vef <= 1.2d+250) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	double t_1 = Math.exp((Vef / KbT)) + 1.0;
	double t_2 = NdChar / t_1;
	double t_3 = NaChar / t_1;
	double tmp;
	if (Vef <= -1.95e+188) {
		tmp = t_3;
	} else if (Vef <= -6.5e+138) {
		tmp = t_2;
	} else if (Vef <= -2.9e-260) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else if (Vef <= 9.8e-286) {
		tmp = t_0;
	} else if (Vef <= 5.6e-137) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else if (Vef <= 5.8e+47) {
		tmp = t_0;
	} else if (Vef <= 1.2e+250) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	t_1 = math.exp((Vef / KbT)) + 1.0
	t_2 = NdChar / t_1
	t_3 = NaChar / t_1
	tmp = 0
	if Vef <= -1.95e+188:
		tmp = t_3
	elif Vef <= -6.5e+138:
		tmp = t_2
	elif Vef <= -2.9e-260:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	elif Vef <= 9.8e-286:
		tmp = t_0
	elif Vef <= 5.6e-137:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	elif Vef <= 5.8e+47:
		tmp = t_0
	elif Vef <= 1.2e+250:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	t_1 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	t_2 = Float64(NdChar / t_1)
	t_3 = Float64(NaChar / t_1)
	tmp = 0.0
	if (Vef <= -1.95e+188)
		tmp = t_3;
	elseif (Vef <= -6.5e+138)
		tmp = t_2;
	elseif (Vef <= -2.9e-260)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	elseif (Vef <= 9.8e-286)
		tmp = t_0;
	elseif (Vef <= 5.6e-137)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (Vef <= 5.8e+47)
		tmp = t_0;
	elseif (Vef <= 1.2e+250)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((EDonor / KbT)) + 1.0);
	t_1 = exp((Vef / KbT)) + 1.0;
	t_2 = NdChar / t_1;
	t_3 = NaChar / t_1;
	tmp = 0.0;
	if (Vef <= -1.95e+188)
		tmp = t_3;
	elseif (Vef <= -6.5e+138)
		tmp = t_2;
	elseif (Vef <= -2.9e-260)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	elseif (Vef <= 9.8e-286)
		tmp = t_0;
	elseif (Vef <= 5.6e-137)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	elseif (Vef <= 5.8e+47)
		tmp = t_0;
	elseif (Vef <= 1.2e+250)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(NdChar / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(NaChar / t$95$1), $MachinePrecision]}, If[LessEqual[Vef, -1.95e+188], t$95$3, If[LessEqual[Vef, -6.5e+138], t$95$2, If[LessEqual[Vef, -2.9e-260], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 9.8e-286], t$95$0, If[LessEqual[Vef, 5.6e-137], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 5.8e+47], t$95$0, If[LessEqual[Vef, 1.2e+250], t$95$3, t$95$2]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if Vef < -1.95e188 or 5.79999999999999961e47 < Vef < 1.20000000000000006e250

   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

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

   5. Simplified 91.6%

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

   6. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 74.4

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

   8. Simplified 74.4%

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

   if -1.95e188 < Vef < -6.50000000000000054e138 or 1.20000000000000006e250 < Vef

   1. Initial program 100.0%

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

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 81.8

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

   5. Simplified 81.8%

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

   6. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 76.4

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

   8. Simplified 76.4%

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

   if -6.50000000000000054e138 < Vef < -2.8999999999999999e-260

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 65.4

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

   5. Simplified 65.4%

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

   6. Taylor expanded in Ev around inf

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

      1. /-lowering-/.f64 42.8

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

   8. Simplified 42.8%

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

   if -2.8999999999999999e-260 < Vef < 9.8000000000000002e-286 or 5.5999999999999998e-137 < Vef < 5.79999999999999961e47

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 75.7

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

   5. Simplified 75.7%

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

   6. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 57.0

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

   8. Simplified 57.0%

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

   if 9.8000000000000002e-286 < Vef < 5.5999999999999998e-137

   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 0

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

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

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 79.4

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

   5. Simplified 79.4%

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

   6. Taylor expanded in EAccept around inf

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

      1. /-lowering-/.f64 48.6

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

   8. Simplified 48.6%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.95 \cdot 10^{+188}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq -6.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq -2.9 \cdot 10^{-260}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 9.8 \cdot 10^{-286}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 5.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 5.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 1.2 \cdot 10^{+250}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -3.1 \cdot 10^{-220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 1.36 \cdot 10^{+62}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)) 1.0))))
   (if (<= NaChar -3.1e-220)
     t_0
     (if (<= NaChar 1.36e+62)
       (/ NdChar (+ (exp (/ (+ (+ Vef EDonor) (- mu Ec)) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -3.1e-220) {
		tmp = t_0;
	} else if (NaChar <= 1.36e+62) {
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp((((ev + eaccept) + (vef - mu)) / kbt)) + 1.0d0)
    if (nachar <= (-3.1d-220)) then
        tmp = t_0
    else if (nachar <= 1.36d+62) then
        tmp = ndchar / (exp((((vef + edonor) + (mu - ec)) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -3.1e-220) {
		tmp = t_0;
	} else if (NaChar <= 1.36e+62) {
		tmp = NdChar / (Math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -3.1e-220:
		tmp = t_0
	elif NaChar <= 1.36e+62:
		tmp = NdChar / (math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -3.1e-220)
		tmp = t_0;
	elseif (NaChar <= 1.36e+62)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Vef + EDonor) + Float64(mu - Ec)) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -3.1e-220)
		tmp = t_0;
	elseif (NaChar <= 1.36e+62)
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -3.1e-220], t$95$0, If[LessEqual[NaChar, 1.36e+62], N[(NdChar / N[(N[Exp[N[(N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -3.10000000000000011e-220 or 1.3600000000000001e62 < 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 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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 71.1

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

   5. Simplified 71.1%

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

   if -3.10000000000000011e-220 < NaChar < 1.3600000000000001e62

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      4. /-lowering-/.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. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right)} + \left(mu + -1 \cdot Ec\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

      14. --lowering--.f64 82.4

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

   5. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu - Ec\right)}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -3.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 1.36 \cdot 10^{+62}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq -3.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;EAccept \leq 7 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept -3.6e-193)
   (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
   (if (<= EAccept 7e+75)
     (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
     (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= -3.6e-193) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else if (EAccept <= 7e+75) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (eaccept <= (-3.6d-193)) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else if (eaccept <= 7d+75) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= -3.6e-193) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else if (EAccept <= 7e+75) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EAccept <= -3.6e-193:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	elif EAccept <= 7e+75:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EAccept <= -3.6e-193)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	elseif (EAccept <= 7e+75)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EAccept <= -3.6e-193)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	elseif (EAccept <= 7e+75)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EAccept, -3.6e-193], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[EAccept, 7e+75], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if EAccept < -3.5999999999999999e-193

   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 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      4. /-lowering-/.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. sub-neg N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef + \color{blue}{-1 \cdot mu}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(EAccept + Ev\right)} + \left(Vef + -1 \cdot mu\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) + \left(Vef + \color{blue}{\left(\mathsf{neg}\left(mu\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) + \color{blue}{\left(Vef - mu\right)}}{KbT}}} \]

      14. --lowering--.f64 65.8

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) + \color{blue}{\left(Vef - mu\right)}}{KbT}}} \]

   5. Simplified 65.8%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}}}} \]

   6. Taylor expanded in Ev around inf

      \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 39.0

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   8. Simplified 39.0%

      \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   if -3.5999999999999999e-193 < EAccept < 6.9999999999999997e75

   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

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{Vef}}{KbT}}} \]
   4. Step-by-step derivation

   5. Simplified 76.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{Vef}}{KbT}}} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{Vef}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{Vef}{KbT}}}} \]

      4. /-lowering-/.f64 51.1

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   8. Simplified 51.1%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}} \]

   if 6.9999999999999997e75 < 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 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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      4. /-lowering-/.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. sub-neg N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \color{blue}{\left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef + \color{blue}{-1 \cdot mu}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(EAccept + Ev\right)} + \left(Vef + -1 \cdot mu\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) + \left(Vef + \color{blue}{\left(\mathsf{neg}\left(mu\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) + \color{blue}{\left(Vef - mu\right)}}{KbT}}} \]

      14. --lowering--.f64 58.4

          \[\leadsto \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) + \color{blue}{\left(Vef - mu\right)}}{KbT}}} \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}}}} \]

   6. Taylor expanded in EAccept around inf

      \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 51.6

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   8. Simplified 51.6%

      \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq -3.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;EAccept \leq 7 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 21.7% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.8 \cdot 10^{-105}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 1.5 \cdot 10^{-155}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -1.8e-105)
   (* NdChar 0.5)
   (if (<= NdChar 1.5e-155) (* 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 <= -1.8e-105) {
		tmp = NdChar * 0.5;
	} else if (NdChar <= 1.5e-155) {
		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 <= (-1.8d-105)) then
        tmp = ndchar * 0.5d0
    else if (ndchar <= 1.5d-155) 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 <= -1.8e-105) {
		tmp = NdChar * 0.5;
	} else if (NdChar <= 1.5e-155) {
		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 <= -1.8e-105:
		tmp = NdChar * 0.5
	elif NdChar <= 1.5e-155:
		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 <= -1.8e-105)
		tmp = Float64(NdChar * 0.5);
	elseif (NdChar <= 1.5e-155)
		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 <= -1.8e-105)
		tmp = NdChar * 0.5;
	elseif (NdChar <= 1.5e-155)
		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, -1.8e-105], N[(NdChar * 0.5), $MachinePrecision], If[LessEqual[NdChar, 1.5e-155], N[(NaChar * 0.5), $MachinePrecision], N[(NdChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.79999999999999982e-105 or 1.49999999999999992e-155 < 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 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 23.4

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 23.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   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. *-lowering-*.f64 20.7

         \[\leadsto \color{blue}{NdChar \cdot 0.5} \]

   8. Simplified 20.7%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} \]

   if -1.79999999999999982e-105 < NdChar < 1.49999999999999992e-155

   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 36.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 36.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   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. *-lowering-*.f64 35.2

         \[\leadsto \color{blue}{NaChar \cdot 0.5} \]

   8. Simplified 35.2%

      \[\leadsto \color{blue}{NaChar \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 27.4% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ \left(NdChar + NaChar\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* (+ NdChar 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 (NdChar + 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 = (ndchar + 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 (NdChar + NaChar) * 0.5;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar + NaChar) * 0.5

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar + NaChar) * 0.5)
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar + NaChar) * 0.5;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar + NaChar), $MachinePrecision] * 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. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   3. +-lowering-+.f64 26.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

5. Simplified 26.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

6. Final simplification 26.7%

   \[\leadsto \left(NdChar + NaChar\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 20: 18.5% 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. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   3. +-lowering-+.f64 26.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

5. Simplified 26.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

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. *-lowering-*.f64 19.1

      \[\leadsto \color{blue}{NaChar \cdot 0.5} \]

8. Simplified 19.1%

   \[\leadsto \color{blue}{NaChar \cdot 0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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))))))