Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 19.1s

Alternatives: 21

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 3.534275563973865e-52
  2. Ec = 1.8096630055632935e+213
  3. Vef = -9.282529722961366e+185
  4. EDonor = -1.5991748693142578e-236
  5. mu = -2.6681421602510446e+104
  6. KbT = 1.4642753930205446e-77
  7. NaChar = -1.231345558957914e-51
  8. Ev = -2.1449187085198105e+162
  9. EAccept = 4.17361469000798e+113

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

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

FPCore

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

Python

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

MATLAB

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

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

Alternative 2: 64.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_2 := t\_1 + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ t_3 := \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+153}:\\ \;\;\;\;t\_1 + \frac{NaChar}{2}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(NdChar, 0.5, t\_3\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+272}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ Vef (- mu Ec))) KbT)))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)))))
        (t_2
         (+
          t_1
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT))))))
        (t_3 (/ NaChar (+ 1.0 (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT))))))
   (if (<= t_2 -5e+153)
     (+ t_1 (/ NaChar 2.0))
     (if (<= t_2 -1e-103)
       (fma NdChar 0.5 t_3)
       (if (<= t_2 5e+22) t_0 (if (<= t_2 5e+272) t_3 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 / (1.0 + exp(((EDonor + (Vef + (mu - Ec))) / KbT)));
	double t_1 = NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)));
	double t_2 = t_1 + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double t_3 = NaChar / (1.0 + exp(((EAccept + (Ev + (Vef - mu))) / KbT)));
	double tmp;
	if (t_2 <= -5e+153) {
		tmp = t_1 + (NaChar / 2.0);
	} else if (t_2 <= -1e-103) {
		tmp = fma(NdChar, 0.5, t_3);
	} else if (t_2 <= 5e+22) {
		tmp = t_0;
	} else if (t_2 <= 5e+272) {
		tmp = t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT))))
	t_2 = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	t_3 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT))))
	tmp = 0.0
	if (t_2 <= -5e+153)
		tmp = Float64(t_1 + Float64(NaChar / 2.0));
	elseif (t_2 <= -1e-103)
		tmp = fma(NdChar, 0.5, t_3);
	elseif (t_2 <= 5e+22)
		tmp = t_0;
	elseif (t_2 <= 5e+272)
		tmp = t_3;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+153], N[(t$95$1 + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-103], N[(NdChar * 0.5 + t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 5e+22], t$95$0, If[LessEqual[t$95$2, 5e+272], t$95$3, 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))))) < -5.00000000000000018e153

   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

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

   5. Simplified 81.5%

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

   if -5.00000000000000018e153 < (+.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))))) < -9.99999999999999958e-104

   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 EDonor around inf

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

      1. lower-/.f64 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in EDonor around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. associate-+r+ N/A

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

      11. mul-1-neg N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 75.2

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

   8. Simplified 75.2%

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

   if -9.99999999999999958e-104 < (+.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 4.99999999999999973e272 < (+.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 inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 81.4%

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

   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))))) < 4.99999999999999973e272

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 70.8

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

   5. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(NdChar, 0.5, \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Vef + EDonor\right) + \left(mu - Ec\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-207}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Vef EDonor) (- mu Ec)))
        (t_1 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_2 -1e-207)
     t_1
     (if (<= t_2 0.0)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- Ec mu) (+ Vef EDonor))) KbT)))
       (if (<= t_2 1e+45)
         t_1
         (if (<= t_2 5e+272)
           (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
           (/ NdChar (+ 1.0 (exp (/ Vef KbT))))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + EDonor) + (mu - Ec);
	double t_1 = NdChar / (1.0 + exp((mu / KbT)));
	double t_2 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_2 <= -1e-207) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((Ec - mu) - (Vef + EDonor))) / KbT));
	} else if (t_2 <= 1e+45) {
		tmp = t_1;
	} else if (t_2 <= 5e+272) {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	} else {
		tmp = NdChar / (1.0 + exp((Vef / KbT)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Vef + EDonor) + Float64(mu - Ec))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT))))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_2 <= -1e-207)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Ec - mu) - Float64(Vef + EDonor))) / KbT)));
	elseif (t_2 <= 1e+45)
		tmp = t_1;
	elseif (t_2 <= 5e+272)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))));
	else
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT))));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-207], t$95$1, If[LessEqual[t$95$2, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Ec - mu), $MachinePrecision] - N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+45], t$95$1, If[LessEqual[t$95$2, 5e+272], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999925e-208 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))))) < 9.9999999999999993e44

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 63.3%

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

   6. Taylor expanded in mu around inf

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

      1. lower-/.f64 44.2

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

   8. Simplified 44.2%

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

   if -9.99999999999999925e-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))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 90.9%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Simplified 87.1%

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

   if 9.9999999999999993e44 < (+.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))))) < 4.99999999999999973e272

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in Ev around inf

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

      1. lower-/.f64 44.5

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

   8. Simplified 44.5%

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

   if 4.99999999999999973e272 < (+.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 inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in Vef around inf

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

      1. lower-/.f64 79.7

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

   8. Simplified 79.7%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-207}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right)}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 10^{+45}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Vef + EDonor\right) + \left(mu - Ec\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_2 \leq -1.6 \cdot 10^{-277}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Vef EDonor) (- mu Ec)))
        (t_1 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_2 -1.6e-277)
     t_1
     (if (<= t_2 0.0)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- Ec mu) (+ Vef EDonor))) KbT)))
       (if (<= t_2 1e+45)
         t_1
         (if (<= t_2 5e+272) (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + EDonor) + (mu - Ec);
	double t_1 = NdChar / (1.0 + exp((EDonor / KbT)));
	double t_2 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_2 <= -1.6e-277) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((Ec - mu) - (Vef + EDonor))) / KbT));
	} else if (t_2 <= 1e+45) {
		tmp = t_1;
	} else if (t_2 <= 5e+272) {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Vef + EDonor) + Float64(mu - Ec))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_2 <= -1.6e-277)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Ec - mu) - Float64(Vef + EDonor))) / KbT)));
	elseif (t_2 <= 1e+45)
		tmp = t_1;
	elseif (t_2 <= 5e+272)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / 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[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.6e-277], t$95$1, If[LessEqual[t$95$2, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Ec - mu), $MachinePrecision] - N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+45], t$95$1, If[LessEqual[t$95$2, 5e+272], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.5999999999999999e-277 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))))) < 9.9999999999999993e44 or 4.99999999999999973e272 < (+.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 inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 65.0%

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

   6. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 42.4

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

   8. Simplified 42.4%

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

   if -1.5999999999999999e-277 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 94.0%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Simplified 89.9%

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

   if 9.9999999999999993e44 < (+.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))))) < 4.99999999999999973e272

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in Ev around inf

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

      1. lower-/.f64 44.5

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

   8. Simplified 44.5%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -1.6 \cdot 10^{-277}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right)}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 10^{+45}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Vef + EDonor\right) + \left(mu - Ec\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-230}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 10^{+133}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{1}{NdChar + NaChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Vef EDonor) (- mu Ec)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_1 -4e-230)
     (* 0.5 (+ NdChar NaChar))
     (if (<= t_1 0.0)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- Ec mu) (+ Vef EDonor))) KbT)))
       (if (<= t_1 1e+133)
         (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))
         (/ 0.5 (/ 1.0 (+ NdChar NaChar))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + EDonor) + (mu - Ec);
	double t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -4e-230) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((Ec - mu) - (Vef + EDonor))) / KbT));
	} else if (t_1 <= 1e+133) {
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	} else {
		tmp = 0.5 / (1.0 / (NdChar + NaChar));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Vef + EDonor) + Float64(mu - Ec))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -4e-230)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Ec - mu) - Float64(Vef + EDonor))) / KbT)));
	elseif (t_1 <= 1e+133)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))));
	else
		tmp = Float64(0.5 / Float64(1.0 / Float64(NdChar + NaChar)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-230], N[(0.5 * N[(NdChar + NaChar), $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[(Ec - mu), $MachinePrecision] - N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+133], N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 / N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.00000000000000019e-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 KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.7

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

   5. Simplified 40.7%

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

   if -4.00000000000000019e-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))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 92.5%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Simplified 88.5%

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

   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))))) < 1e133

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 46.2

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

   5. Simplified 46.2%

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

   6. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 29.4

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

   8. Simplified 29.4%

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

   if 1e133 < (+.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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 41.5

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

   5. Simplified 41.5%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 9.7

          \[\leadsto 0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \color{blue}{\left(NdChar - NaChar\right)}}} \]

   7. Applied egg-rr 9.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 9.7

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares N/A

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

      15. lift--.f64 N/A

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

      16. flip-+ N/A

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

      17. lift-+.f64 N/A

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

      18. lower-/.f64 41.5

          \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{NdChar + NaChar}}} \]

   9. Applied egg-rr 41.5%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{NdChar + NaChar}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -4 \cdot 10^{-230}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right)}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 10^{+133}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{1}{NdChar + NaChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ t_1 := t\_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + t\_0\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-112}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))
        (t_1 (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          t_0)))
   (if (<= t_2 -1e-303)
     t_1
     (if (<= t_2 1e-112)
       (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ Vef (- mu Ec))) 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 = NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	double t_2 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + t_0;
	double tmp;
	if (t_2 <= -1e-303) {
		tmp = t_1;
	} else if (t_2 <= 1e-112) {
		tmp = NdChar / (1.0 + exp(((EDonor + (Vef + (mu - Ec))) / KbT)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((((vef + ev) + eaccept) - mu) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    t_2 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + t_0
    if (t_2 <= (-1d-303)) then
        tmp = t_1
    else if (t_2 <= 1d-112) then
        tmp = ndchar / (1.0d0 + exp(((edonor + (vef + (mu - ec))) / kbt)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	double t_2 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + t_0;
	double tmp;
	if (t_2 <= -1e-303) {
		tmp = t_1;
	} else if (t_2 <= 1e-112) {
		tmp = NdChar / (1.0 + Math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)))
	t_1 = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	t_2 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + t_0
	tmp = 0
	if t_2 <= -1e-303:
		tmp = t_1
	elif t_2 <= 1e-112:
		tmp = NdChar / (1.0 + math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + t_0)
	tmp = 0.0
	if (t_2 <= -1e-303)
		tmp = t_1;
	elseif (t_2 <= 1e-112)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT)));
	t_1 = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	t_2 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + t_0;
	tmp = 0.0;
	if (t_2 <= -1e-303)
		tmp = t_1;
	elseif (t_2 <= 1e-112)
		tmp = NdChar / (1.0 + exp(((EDonor + (Vef + (mu - Ec))) / KbT)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-303], t$95$1, If[LessEqual[t$95$2, 1e-112], N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $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))))) < -9.99999999999999931e-304 or 9.9999999999999995e-113 < (+.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 EDonor around inf

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

      1. lower-/.f64 79.9

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

   5. Simplified 79.9%

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

   if -9.99999999999999931e-304 < (+.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-113

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 87.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 10^{-112}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Vef + EDonor\right) + \left(mu - Ec\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-230}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-300}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Vef EDonor) (- mu Ec)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_1 -4e-230)
     (* 0.5 (+ NdChar NaChar))
     (if (<= t_1 1e-300)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- Ec mu) (+ Vef EDonor))) KbT)))
       (/ NaChar (+ 1.0 (exp (/ Ev KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + EDonor) + (mu - Ec);
	double t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -4e-230) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (t_1 <= 1e-300) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((Ec - mu) - (Vef + EDonor))) / KbT));
	} else {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Vef + EDonor) + Float64(mu - Ec))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -4e-230)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	elseif (t_1 <= 1e-300)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Ec - mu) - Float64(Vef + EDonor))) / KbT)));
	else
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-230], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-300], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Ec - mu), $MachinePrecision] - N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.00000000000000019e-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 KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.7

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

   5. Simplified 40.7%

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

   if -4.00000000000000019e-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))))) < 1.00000000000000003e-300

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 92.6%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Simplified 86.9%

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

   if 1.00000000000000003e-300 < (+.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. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 50.2

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

   5. Simplified 50.2%

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

   6. Taylor expanded in Ev around inf

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

      1. lower-/.f64 31.0

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

   8. Simplified 31.0%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -4 \cdot 10^{-230}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 10^{-300}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right)}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Vef + EDonor\right) + \left(mu - Ec\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-230}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-267}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Vef EDonor) (- mu Ec)))
        (t_1 (* 0.5 (+ NdChar NaChar)))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_2 -4e-230)
     t_1
     (if (<= t_2 5e-267)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- Ec mu) (+ Vef EDonor))) KbT)))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + EDonor) + (mu - Ec);
	double t_1 = 0.5 * (NdChar + NaChar);
	double t_2 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_2 <= -4e-230) {
		tmp = t_1;
	} else if (t_2 <= 5e-267) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((Ec - mu) - (Vef + EDonor))) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Vef + EDonor) + Float64(mu - Ec))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_2 <= -4e-230)
		tmp = t_1;
	elseif (t_2 <= 5e-267)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Ec - mu) - Float64(Vef + EDonor))) / KbT)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-230], t$95$1, If[LessEqual[t$95$2, 5e-267], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Ec - mu), $MachinePrecision] - N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.00000000000000019e-230 or 4.9999999999999999e-267 < (+.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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 34.3

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

   5. Simplified 34.3%

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

   if -4.00000000000000019e-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))))) < 4.9999999999999999e-267

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 91.3%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   8. Simplified 82.4%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -4 \cdot 10^{-230}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-267}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right) \cdot \left(\left(Vef + EDonor\right) + \left(mu - Ec\right)\right)}{KbT}, \left(Ec - mu\right) - \left(Vef + EDonor\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-292}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\frac{EDonor}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_1 -1e-85)
     t_0
     (if (<= t_1 1e-292)
       (/
        NdChar
        (- (+ (+ (/ EDonor KbT) 2.0) (+ (/ Vef KbT) (/ mu KbT))) (/ Ec KbT)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-85) {
		tmp = t_0;
	} else if (t_1 <= 1e-292) {
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp(((((vef + ev) + eaccept) - mu) / kbt))))
    if (t_1 <= (-1d-85)) then
        tmp = t_0
    else if (t_1 <= 1d-292) then
        tmp = ndchar / ((((edonor / kbt) + 2.0d0) + ((vef / kbt) + (mu / kbt))) - (ec / kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-85) {
		tmp = t_0;
	} else if (t_1 <= 1e-292) {
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -1e-85:
		tmp = t_0
	elif t_1 <= 1e-292:
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -1e-85)
		tmp = t_0;
	elseif (t_1 <= 1e-292)
		tmp = Float64(NdChar / Float64(Float64(Float64(Float64(EDonor / KbT) + 2.0) + Float64(Float64(Vef / KbT) + Float64(mu / KbT))) - Float64(Ec / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -1e-85)
		tmp = t_0;
	elseif (t_1 <= 1e-292)
		tmp = NdChar / ((((EDonor / KbT) + 2.0) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-85], t$95$0, If[LessEqual[t$95$1, 1e-292], N[(NdChar / N[(N[(N[(N[(EDonor / KbT), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999998e-86 or 1.0000000000000001e-292 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 34.4

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

   5. Simplified 34.4%

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

   if -9.9999999999999998e-86 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.0000000000000001e-292

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 87.5%

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

   6. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 44.5

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

   8. Simplified 44.5%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 10^{-292}:\\ \;\;\;\;\frac{NdChar}{\left(\left(\frac{EDonor}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-234}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\frac{\frac{NdChar \cdot NdChar}{NaChar} - NdChar}{NaChar} - -1}{NaChar}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_1 -2e-259)
     t_0
     (if (<= t_1 5e-234)
       (*
        0.5
        (/
         1.0
         (/
          (- (/ (- (/ (* NdChar NdChar) NaChar) NdChar) NaChar) -1.0)
          NaChar)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 5e-234) {
		tmp = 0.5 * (1.0 / ((((((NdChar * NdChar) / NaChar) - NdChar) / NaChar) - -1.0) / NaChar));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp(((((vef + ev) + eaccept) - mu) / kbt))))
    if (t_1 <= (-2d-259)) then
        tmp = t_0
    else if (t_1 <= 5d-234) then
        tmp = 0.5d0 * (1.0d0 / ((((((ndchar * ndchar) / nachar) - ndchar) / nachar) - (-1.0d0)) / nachar))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 5e-234) {
		tmp = 0.5 * (1.0 / ((((((NdChar * NdChar) / NaChar) - NdChar) / NaChar) - -1.0) / NaChar));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -2e-259:
		tmp = t_0
	elif t_1 <= 5e-234:
		tmp = 0.5 * (1.0 / ((((((NdChar * NdChar) / NaChar) - NdChar) / NaChar) - -1.0) / NaChar))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 5e-234)
		tmp = Float64(0.5 * Float64(1.0 / Float64(Float64(Float64(Float64(Float64(Float64(NdChar * NdChar) / NaChar) - NdChar) / NaChar) - -1.0) / NaChar)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 5e-234)
		tmp = 0.5 * (1.0 / ((((((NdChar * NdChar) / NaChar) - NdChar) / NaChar) - -1.0) / NaChar));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-259], t$95$0, If[LessEqual[t$95$1, 5e-234], N[(0.5 * N[(1.0 / N[(N[(N[(N[(N[(N[(NdChar * NdChar), $MachinePrecision] / NaChar), $MachinePrecision] - NdChar), $MachinePrecision] / NaChar), $MachinePrecision] - -1.0), $MachinePrecision] / NaChar), $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))))) < -2.0000000000000001e-259 or 4.99999999999999979e-234 < (+.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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 35.0

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

   5. Simplified 35.0%

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

   if -2.0000000000000001e-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))))) < 4.99999999999999979e-234

   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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 3.2

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

   5. Simplified 3.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 4.7

          \[\leadsto 0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \color{blue}{\left(NdChar - NaChar\right)}}} \]

   7. Applied egg-rr 4.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NaChar around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 43.8

          \[\leadsto 0.5 \cdot \frac{1}{\frac{\frac{\frac{NdChar \cdot NdChar}{NaChar} - NdChar}{-NaChar} + -1}{\color{blue}{-NaChar}}} \]

   10. Simplified 43.8%

       \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{\frac{\frac{NdChar \cdot NdChar}{NaChar} - NdChar}{-NaChar} + -1}{-NaChar}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-234}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{\frac{\frac{NdChar \cdot NdChar}{NaChar} - NdChar}{NaChar} - -1}{NaChar}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\frac{\frac{0.5}{NdChar - NaChar}}{\frac{-1}{NaChar \cdot NaChar}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_1 -2e-259)
     t_0
     (if (<= t_1 5e-293)
       (/ (/ 0.5 (- NdChar NaChar)) (/ -1.0 (* NaChar NaChar)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 5e-293) {
		tmp = (0.5 / (NdChar - NaChar)) / (-1.0 / (NaChar * NaChar));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp(((((vef + ev) + eaccept) - mu) / kbt))))
    if (t_1 <= (-2d-259)) then
        tmp = t_0
    else if (t_1 <= 5d-293) then
        tmp = (0.5d0 / (ndchar - nachar)) / ((-1.0d0) / (nachar * nachar))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 5e-293) {
		tmp = (0.5 / (NdChar - NaChar)) / (-1.0 / (NaChar * NaChar));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -2e-259:
		tmp = t_0
	elif t_1 <= 5e-293:
		tmp = (0.5 / (NdChar - NaChar)) / (-1.0 / (NaChar * NaChar))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 5e-293)
		tmp = Float64(Float64(0.5 / Float64(NdChar - NaChar)) / Float64(-1.0 / Float64(NaChar * NaChar)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 5e-293)
		tmp = (0.5 / (NdChar - NaChar)) / (-1.0 / (NaChar * NaChar));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-259], t$95$0, If[LessEqual[t$95$1, 5e-293], N[(N[(0.5 / N[(NdChar - NaChar), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[(NaChar * NaChar), $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))))) < -2.0000000000000001e-259 or 5.0000000000000003e-293 < (+.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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 34.1

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

   5. Simplified 34.1%

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

   if -2.0000000000000001e-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))))) < 5.0000000000000003e-293

   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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 2.9

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

   5. Simplified 2.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 4.9

          \[\leadsto 0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \color{blue}{\left(NdChar - NaChar\right)}}} \]

   7. Applied egg-rr 4.9%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 4.9

          \[\leadsto \frac{\frac{0.5}{NdChar - NaChar}}{\color{blue}{\frac{1}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 4.9

          \[\leadsto \frac{\frac{0.5}{NdChar - NaChar}}{\frac{1}{\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NdChar - NaChar\right)}} \]

   9. Applied egg-rr 4.9%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{NdChar - NaChar}}{\frac{1}{\left(NdChar + NaChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   10. Taylor expanded in NdChar around 0

       \[\leadsto \frac{\frac{\frac{1}{2}}{NdChar - NaChar}}{\color{blue}{\frac{-1}{{NaChar}^{2}}}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{NdChar - NaChar}}{\color{blue}{\frac{-1}{{NaChar}^{2}}}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 38.1

          \[\leadsto \frac{\frac{0.5}{NdChar - NaChar}}{\frac{-1}{\color{blue}{NaChar \cdot NaChar}}} \]

   12. Simplified 38.1%

       \[\leadsto \frac{\frac{0.5}{NdChar - NaChar}}{\color{blue}{\frac{-1}{NaChar \cdot NaChar}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\frac{\frac{0.5}{NdChar - NaChar}}{\frac{-1}{NaChar \cdot NaChar}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-293}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{NaChar - NdChar}{NaChar \cdot NaChar}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_1 -2e-259)
     t_0
     (if (<= t_1 5e-293)
       (* 0.5 (/ 1.0 (/ (- NaChar NdChar) (* NaChar NaChar))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 5e-293) {
		tmp = 0.5 * (1.0 / ((NaChar - NdChar) / (NaChar * NaChar)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp(((((vef + ev) + eaccept) - mu) / kbt))))
    if (t_1 <= (-2d-259)) then
        tmp = t_0
    else if (t_1 <= 5d-293) then
        tmp = 0.5d0 * (1.0d0 / ((nachar - ndchar) / (nachar * nachar)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 5e-293) {
		tmp = 0.5 * (1.0 / ((NaChar - NdChar) / (NaChar * NaChar)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -2e-259:
		tmp = t_0
	elif t_1 <= 5e-293:
		tmp = 0.5 * (1.0 / ((NaChar - NdChar) / (NaChar * NaChar)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 5e-293)
		tmp = Float64(0.5 * Float64(1.0 / Float64(Float64(NaChar - NdChar) / Float64(NaChar * NaChar))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 5e-293)
		tmp = 0.5 * (1.0 / ((NaChar - NdChar) / (NaChar * NaChar)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-259], t$95$0, If[LessEqual[t$95$1, 5e-293], N[(0.5 * N[(1.0 / N[(N[(NaChar - NdChar), $MachinePrecision] / N[(NaChar * NaChar), $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))))) < -2.0000000000000001e-259 or 5.0000000000000003e-293 < (+.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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 34.1

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

   5. Simplified 34.1%

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

   if -2.0000000000000001e-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))))) < 5.0000000000000003e-293

   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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 2.9

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

   5. Simplified 2.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 4.9

          \[\leadsto 0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \color{blue}{\left(NdChar - NaChar\right)}}} \]

   7. Applied egg-rr 4.9%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NaChar around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 36.2

         \[\leadsto 0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{NaChar \cdot \color{blue}{\left(-NaChar\right)}}} \]

   10. Simplified 36.2%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-293}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{NaChar - NdChar}{NaChar \cdot NaChar}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{NdChar \cdot NdChar}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_1 -2e-259)
     t_0
     (if (<= t_1 5e-267)
       (* 0.5 (/ 1.0 (/ (- NdChar NaChar) (* NdChar NdChar))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 5e-267) {
		tmp = 0.5 * (1.0 / ((NdChar - NaChar) / (NdChar * NdChar)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp(((((vef + ev) + eaccept) - mu) / kbt))))
    if (t_1 <= (-2d-259)) then
        tmp = t_0
    else if (t_1 <= 5d-267) then
        tmp = 0.5d0 * (1.0d0 / ((ndchar - nachar) / (ndchar * ndchar)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 5e-267) {
		tmp = 0.5 * (1.0 / ((NdChar - NaChar) / (NdChar * NdChar)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -2e-259:
		tmp = t_0
	elif t_1 <= 5e-267:
		tmp = 0.5 * (1.0 / ((NdChar - NaChar) / (NdChar * NdChar)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 5e-267)
		tmp = Float64(0.5 * Float64(1.0 / Float64(Float64(NdChar - NaChar) / Float64(NdChar * NdChar))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 5e-267)
		tmp = 0.5 * (1.0 / ((NdChar - NaChar) / (NdChar * NdChar)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-259], t$95$0, If[LessEqual[t$95$1, 5e-267], N[(0.5 * N[(1.0 / N[(N[(NdChar - NaChar), $MachinePrecision] / N[(NdChar * NdChar), $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))))) < -2.0000000000000001e-259 or 4.9999999999999999e-267 < (+.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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 34.4

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

   5. Simplified 34.4%

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

   if -2.0000000000000001e-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))))) < 4.9999999999999999e-267

   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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 2.9

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

   5. Simplified 2.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 4.8

          \[\leadsto 0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \color{blue}{\left(NdChar - NaChar\right)}}} \]

   7. Applied egg-rr 4.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NaChar around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 28.7

         \[\leadsto 0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{\color{blue}{NdChar \cdot NdChar}}} \]

   10. Simplified 28.7%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{\color{blue}{NdChar \cdot NdChar}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{NdChar \cdot NdChar}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-292}:\\ \;\;\;\;-0.25 \cdot \frac{NdChar \cdot EDonor}{KbT}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)))))))
   (if (<= t_1 -1e-303)
     t_0
     (if (<= t_1 1e-292) (* -0.25 (/ (* NdChar EDonor) KbT)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-303) {
		tmp = t_0;
	} else if (t_1 <= 1e-292) {
		tmp = -0.25 * ((NdChar * EDonor) / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp(((((vef + ev) + eaccept) - mu) / kbt))))
    if (t_1 <= (-1d-303)) then
        tmp = t_0
    else if (t_1 <= 1d-292) then
        tmp = (-0.25d0) * ((ndchar * edonor) / kbt)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-303) {
		tmp = t_0;
	} else if (t_1 <= 1e-292) {
		tmp = -0.25 * ((NdChar * EDonor) / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp(((((Vef + Ev) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -1e-303:
		tmp = t_0
	elif t_1 <= 1e-292:
		tmp = -0.25 * ((NdChar * EDonor) / KbT)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -1e-303)
		tmp = t_0;
	elseif (t_1 <= 1e-292)
		tmp = Float64(-0.25 * Float64(Float64(NdChar * EDonor) / KbT));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((((Vef + Ev) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -1e-303)
		tmp = t_0;
	elseif (t_1 <= 1e-292)
		tmp = -0.25 * ((NdChar * EDonor) / KbT);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-303], t$95$0, If[LessEqual[t$95$1, 1e-292], N[(-0.25 * N[(N[(NdChar * EDonor), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999931e-304 or 1.0000000000000001e-292 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 33.7

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

   5. Simplified 33.7%

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

   if -9.99999999999999931e-304 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.0000000000000001e-292

   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 1.4%

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

   6. Taylor expanded in EDonor around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{EDonor \cdot NdChar}{KbT}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{EDonor \cdot NdChar}{KbT}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{EDonor \cdot NdChar}{KbT}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\color{blue}{NdChar \cdot EDonor}}{KbT} \]

      4. lower-*.f64 24.4

         \[\leadsto -0.25 \cdot \frac{\color{blue}{NdChar \cdot EDonor}}{KbT} \]

   8. Simplified 24.4%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{NdChar \cdot EDonor}{KbT}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}}} \leq 10^{-292}:\\ \;\;\;\;-0.25 \cdot \frac{NdChar \cdot EDonor}{KbT}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -1.15 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq -9.4 \cdot 10^{-234}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 3.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT))))))
   (if (<= NaChar -1.15e-120)
     t_0
     (if (<= NaChar -9.4e-234)
       (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
       (if (<= NaChar 3.6e-147) (/ NdChar (+ 1.0 (exp (/ 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 = NaChar / (1.0 + exp(((EAccept + (Ev + (Vef - mu))) / KbT)));
	double tmp;
	if (NaChar <= -1.15e-120) {
		tmp = t_0;
	} else if (NaChar <= -9.4e-234) {
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	} else if (NaChar <= 3.6e-147) {
		tmp = NdChar / (1.0 + exp((Vef / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((eaccept + (ev + (vef - mu))) / kbt)))
    if (nachar <= (-1.15d-120)) then
        tmp = t_0
    else if (nachar <= (-9.4d-234)) then
        tmp = ndchar / (1.0d0 + exp((edonor / kbt)))
    else if (nachar <= 3.6d-147) then
        tmp = ndchar / (1.0d0 + exp((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 = NaChar / (1.0 + Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)));
	double tmp;
	if (NaChar <= -1.15e-120) {
		tmp = t_0;
	} else if (NaChar <= -9.4e-234) {
		tmp = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	} else if (NaChar <= 3.6e-147) {
		tmp = NdChar / (1.0 + Math.exp((Vef / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)))
	tmp = 0
	if NaChar <= -1.15e-120:
		tmp = t_0
	elif NaChar <= -9.4e-234:
		tmp = NdChar / (1.0 + math.exp((EDonor / KbT)))
	elif NaChar <= 3.6e-147:
		tmp = NdChar / (1.0 + math.exp((Vef / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT))))
	tmp = 0.0
	if (NaChar <= -1.15e-120)
		tmp = t_0;
	elseif (NaChar <= -9.4e-234)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))));
	elseif (NaChar <= 3.6e-147)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((EAccept + (Ev + (Vef - mu))) / KbT)));
	tmp = 0.0;
	if (NaChar <= -1.15e-120)
		tmp = t_0;
	elseif (NaChar <= -9.4e-234)
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	elseif (NaChar <= 3.6e-147)
		tmp = NdChar / (1.0 + exp((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[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.15e-120], t$95$0, If[LessEqual[NaChar, -9.4e-234], N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 3.6e-147], N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -1.14999999999999993e-120 or 3.60000000000000012e-147 < 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. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 61.7

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

   5. Simplified 61.7%

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

   if -1.14999999999999993e-120 < NaChar < -9.4000000000000002e-234

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 91.4%

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

   6. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 71.5

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

   8. Simplified 71.5%

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

   if -9.4000000000000002e-234 < NaChar < 3.60000000000000012e-147

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 83.2%

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

   6. Taylor expanded in Vef around inf

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

      1. lower-/.f64 61.1

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

   8. Simplified 61.1%

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

Alternative 16: 41.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -5.2 \cdot 10^{+261}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq -3.95 \cdot 10^{+186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq -5.1 \cdot 10^{-231}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT))))))
   (if (<= mu -5.2e+261)
     (/ NaChar (+ 1.0 (exp (/ mu (- KbT)))))
     (if (<= mu -3.95e+186)
       t_0
       (if (<= mu -5.1e-231)
         (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
         (if (<= mu 1.3e+32) (/ NdChar (+ 1.0 (exp (/ 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 / (1.0 + exp((mu / KbT)));
	double tmp;
	if (mu <= -5.2e+261) {
		tmp = NaChar / (1.0 + exp((mu / -KbT)));
	} else if (mu <= -3.95e+186) {
		tmp = t_0;
	} else if (mu <= -5.1e-231) {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	} else if (mu <= 1.3e+32) {
		tmp = NdChar / (1.0 + exp((Vef / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((mu / kbt)))
    if (mu <= (-5.2d+261)) then
        tmp = nachar / (1.0d0 + exp((mu / -kbt)))
    else if (mu <= (-3.95d+186)) then
        tmp = t_0
    else if (mu <= (-5.1d-231)) then
        tmp = nachar / (1.0d0 + exp((ev / kbt)))
    else if (mu <= 1.3d+32) then
        tmp = ndchar / (1.0d0 + exp((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 / (1.0 + Math.exp((mu / KbT)));
	double tmp;
	if (mu <= -5.2e+261) {
		tmp = NaChar / (1.0 + Math.exp((mu / -KbT)));
	} else if (mu <= -3.95e+186) {
		tmp = t_0;
	} else if (mu <= -5.1e-231) {
		tmp = NaChar / (1.0 + Math.exp((Ev / KbT)));
	} else if (mu <= 1.3e+32) {
		tmp = NdChar / (1.0 + Math.exp((Vef / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((mu / KbT)))
	tmp = 0
	if mu <= -5.2e+261:
		tmp = NaChar / (1.0 + math.exp((mu / -KbT)))
	elif mu <= -3.95e+186:
		tmp = t_0
	elif mu <= -5.1e-231:
		tmp = NaChar / (1.0 + math.exp((Ev / KbT)))
	elif mu <= 1.3e+32:
		tmp = NdChar / (1.0 + math.exp((Vef / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT))))
	tmp = 0.0
	if (mu <= -5.2e+261)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT)))));
	elseif (mu <= -3.95e+186)
		tmp = t_0;
	elseif (mu <= -5.1e-231)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))));
	elseif (mu <= 1.3e+32)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((mu / KbT)));
	tmp = 0.0;
	if (mu <= -5.2e+261)
		tmp = NaChar / (1.0 + exp((mu / -KbT)));
	elseif (mu <= -3.95e+186)
		tmp = t_0;
	elseif (mu <= -5.1e-231)
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	elseif (mu <= 1.3e+32)
		tmp = NdChar / (1.0 + exp((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[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -5.2e+261], N[(NaChar / N[(1.0 + N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, -3.95e+186], t$95$0, If[LessEqual[mu, -5.1e-231], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.3e+32], N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -5.19999999999999963e261

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 91.1

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

   5. Simplified 91.1%

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

   6. Taylor expanded in mu around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.1

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

   8. Simplified 91.1%

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

   if -5.19999999999999963e261 < mu < -3.95000000000000001e186 or 1.3000000000000001e32 < mu

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 77.6%

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

   6. Taylor expanded in mu around inf

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

      1. lower-/.f64 65.1

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

   8. Simplified 65.1%

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

   if -3.95000000000000001e186 < mu < -5.1e-231

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 65.7

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

   5. Simplified 65.7%

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

   6. Taylor expanded in Ev around inf

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

      1. lower-/.f64 47.6

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

   8. Simplified 47.6%

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

   if -5.1e-231 < mu < 1.3000000000000001e32

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 69.5%

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

   6. Taylor expanded in Vef around inf

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

      1. lower-/.f64 52.9

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

   8. Simplified 52.9%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -5.2 \cdot 10^{+261}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq -3.95 \cdot 10^{+186}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -5.1 \cdot 10^{-231}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -0.27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 4.9 \cdot 10^{-194}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ Vef (- mu Ec))) KbT))))))
   (if (<= NdChar -0.27)
     t_0
     (if (<= NdChar 4.9e-194)
       (/ NaChar (+ 1.0 (exp (/ (+ EAccept (+ Ev (- Vef 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 / (1.0 + exp(((EDonor + (Vef + (mu - Ec))) / KbT)));
	double tmp;
	if (NdChar <= -0.27) {
		tmp = t_0;
	} else if (NdChar <= 4.9e-194) {
		tmp = NaChar / (1.0 + exp(((EAccept + (Ev + (Vef - mu))) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((edonor + (vef + (mu - ec))) / kbt)))
    if (ndchar <= (-0.27d0)) then
        tmp = t_0
    else if (ndchar <= 4.9d-194) then
        tmp = nachar / (1.0d0 + exp(((eaccept + (ev + (vef - 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 / (1.0 + Math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)));
	double tmp;
	if (NdChar <= -0.27) {
		tmp = t_0;
	} else if (NdChar <= 4.9e-194) {
		tmp = NaChar / (1.0 + Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)))
	tmp = 0
	if NdChar <= -0.27:
		tmp = t_0
	elif NdChar <= 4.9e-194:
		tmp = NaChar / (1.0 + math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT))))
	tmp = 0.0
	if (NdChar <= -0.27)
		tmp = t_0;
	elseif (NdChar <= 4.9e-194)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - 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 / (1.0 + exp(((EDonor + (Vef + (mu - Ec))) / KbT)));
	tmp = 0.0;
	if (NdChar <= -0.27)
		tmp = t_0;
	elseif (NdChar <= 4.9e-194)
		tmp = NaChar / (1.0 + exp(((EAccept + (Ev + (Vef - 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[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -0.27], t$95$0, If[LessEqual[NdChar, 4.9e-194], N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -0.27000000000000002 or 4.90000000000000004e-194 < NdChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 73.3%

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

   if -0.27000000000000002 < NdChar < 4.90000000000000004e-194

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 74.3

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

   5. Simplified 74.3%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -0.27:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 4.9 \cdot 10^{-194}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\frac{Vef}{KbT}}\\ \mathbf{if}\;NaChar \leq -1.55 \cdot 10^{-56}:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{elif}\;NaChar \leq -3.1 \cdot 10^{-236}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.65 \cdot 10^{+68}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (/ Vef KbT)))))
   (if (<= NaChar -1.55e-56)
     (/ NaChar t_0)
     (if (<= NaChar -3.1e-236)
       (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
       (if (<= NaChar 1.65e+68)
         (/ NdChar t_0)
         (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 1.0 + exp((Vef / KbT));
	double tmp;
	if (NaChar <= -1.55e-56) {
		tmp = NaChar / t_0;
	} else if (NaChar <= -3.1e-236) {
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	} else if (NaChar <= 1.65e+68) {
		tmp = NdChar / t_0;
	} else {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + exp((vef / kbt))
    if (nachar <= (-1.55d-56)) then
        tmp = nachar / t_0
    else if (nachar <= (-3.1d-236)) then
        tmp = ndchar / (1.0d0 + exp((edonor / kbt)))
    else if (nachar <= 1.65d+68) then
        tmp = ndchar / t_0
    else
        tmp = nachar / (1.0d0 + exp((ev / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 1.0 + Math.exp((Vef / KbT));
	double tmp;
	if (NaChar <= -1.55e-56) {
		tmp = NaChar / t_0;
	} else if (NaChar <= -3.1e-236) {
		tmp = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	} else if (NaChar <= 1.65e+68) {
		tmp = NdChar / t_0;
	} else {
		tmp = NaChar / (1.0 + Math.exp((Ev / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 1.0 + math.exp((Vef / KbT))
	tmp = 0
	if NaChar <= -1.55e-56:
		tmp = NaChar / t_0
	elif NaChar <= -3.1e-236:
		tmp = NdChar / (1.0 + math.exp((EDonor / KbT)))
	elif NaChar <= 1.65e+68:
		tmp = NdChar / t_0
	else:
		tmp = NaChar / (1.0 + math.exp((Ev / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(1.0 + exp(Float64(Vef / KbT)))
	tmp = 0.0
	if (NaChar <= -1.55e-56)
		tmp = Float64(NaChar / t_0);
	elseif (NaChar <= -3.1e-236)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))));
	elseif (NaChar <= 1.65e+68)
		tmp = Float64(NdChar / t_0);
	else
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 1.0 + exp((Vef / KbT));
	tmp = 0.0;
	if (NaChar <= -1.55e-56)
		tmp = NaChar / t_0;
	elseif (NaChar <= -3.1e-236)
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	elseif (NaChar <= 1.65e+68)
		tmp = NdChar / t_0;
	else
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.55e-56], N[(NaChar / t$95$0), $MachinePrecision], If[LessEqual[NaChar, -3.1e-236], N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.65e+68], N[(NdChar / t$95$0), $MachinePrecision], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -1.54999999999999994e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 61.9

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

   5. Simplified 61.9%

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

   6. Taylor expanded in Vef around inf

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

      1. lower-/.f64 45.9

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

   8. Simplified 45.9%

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

   if -1.54999999999999994e-56 < NaChar < -3.0999999999999998e-236

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 79.9%

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

   6. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 56.3

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

   8. Simplified 56.3%

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

   if -3.0999999999999998e-236 < NaChar < 1.65e68

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 75.4%

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

   6. Taylor expanded in Vef around inf

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

      1. lower-/.f64 55.6

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

   8. Simplified 55.6%

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

   if 1.65e68 < 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. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 68.4

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

   5. Simplified 68.4%

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

   6. Taylor expanded in Ev around inf

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

      1. lower-/.f64 42.0

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

   8. Simplified 42.0%

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

Alternative 19: 23.1% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -1.3 \cdot 10^{-105}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq 6 \cdot 10^{+84}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -1.3e-105)
   (* NaChar 0.5)
   (if (<= NaChar 6e+84) (* NdChar 0.5) (* NaChar 0.5))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.3e-105) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 6e+84) {
		tmp = NdChar * 0.5;
	} else {
		tmp = NaChar * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-1.3d-105)) then
        tmp = nachar * 0.5d0
    else if (nachar <= 6d+84) then
        tmp = ndchar * 0.5d0
    else
        tmp = nachar * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.3e-105) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 6e+84) {
		tmp = NdChar * 0.5;
	} else {
		tmp = NaChar * 0.5;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -1.3e-105:
		tmp = NaChar * 0.5
	elif NaChar <= 6e+84:
		tmp = NdChar * 0.5
	else:
		tmp = NaChar * 0.5
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -1.3e-105)
		tmp = Float64(NaChar * 0.5);
	elseif (NaChar <= 6e+84)
		tmp = Float64(NdChar * 0.5);
	else
		tmp = Float64(NaChar * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -1.3e-105)
		tmp = NaChar * 0.5;
	elseif (NaChar <= 6e+84)
		tmp = NdChar * 0.5;
	else
		tmp = NaChar * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -1.3e-105], N[(NaChar * 0.5), $MachinePrecision], If[LessEqual[NaChar, 6e+84], N[(NdChar * 0.5), $MachinePrecision], N[(NaChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.2999999999999999e-105 or 5.99999999999999992e84 < 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 KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 25.7

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

   5. Simplified 25.7%

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

   6. Taylor expanded in NdChar around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 23.0

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

   8. Simplified 23.0%

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

   if -1.2999999999999999e-105 < NaChar < 5.99999999999999992e84

   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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 31.0

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

   5. Simplified 31.0%

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

   6. Taylor expanded in NdChar around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.4

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

   8. Simplified 30.4%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 28.6% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(NdChar + NaChar\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 28.3

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

5. Simplified 28.3%

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

Alternative 21: 18.1% 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. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 28.3

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

5. Simplified 28.3%

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

6. Taylor expanded in NdChar around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 16.5

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

8. Simplified 16.5%

   \[\leadsto \color{blue}{NaChar \cdot 0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))))