Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 19.7s

Alternatives: 24

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = -5.4485266002971924e-219
  2. Ec = -1.571894543251409e-192
  3. Vef = -5.266711140129765e-230
  4. EDonor = 3.086359105619781e+186
  5. mu = 7.073034306733683e-222
  6. KbT = -9.844126922296806e+22
  7. NaChar = -9.675566343278425e-219
  8. Ev = -5.044715222686643e-303
  9. EAccept = 5.6334455053251864e+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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 46.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_2 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_2 (+ EDonor (+ Vef (- mu Ec)))))
   (if (<= t_1 -5e-50)
     (* (- NaChar NdChar) (* (+ NdChar NaChar) (/ 0.5 (- NaChar NdChar))))
     (if (<= t_1 -2e-291)
       (/ NdChar (+ (exp (/ mu KbT)) 1.0))
       (if (<= t_1 5e-270)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_2 t_2) KbT)
             (* Ec (- 1.0 (/ (+ EDonor (+ Vef mu)) Ec))))
            KbT)))
         (if (<= t_1 1e+27)
           t_0
           (if (<= t_1 4e+191)
             (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
             t_0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_2 = EDonor + (Vef + (mu - Ec));
	double tmp;
	if (t_1 <= -5e-50) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_1 <= -2e-291) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else if (t_1 <= 5e-270) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_2 * t_2) / KbT), (Ec * (1.0 - ((EDonor + (Vef + mu)) / Ec)))) / KbT));
	} else if (t_1 <= 1e+27) {
		tmp = t_0;
	} else if (t_1 <= 4e+191) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_2 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	tmp = 0.0
	if (t_1 <= -5e-50)
		tmp = Float64(Float64(NaChar - NdChar) * Float64(Float64(NdChar + NaChar) * Float64(0.5 / Float64(NaChar - NdChar))));
	elseif (t_1 <= -2e-291)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	elseif (t_1 <= 5e-270)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Float64(Ec * Float64(1.0 - Float64(Float64(EDonor + Float64(Vef + mu)) / Ec)))) / KbT)));
	elseif (t_1 <= 1e+27)
		tmp = t_0;
	elseif (t_1 <= 4e+191)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-50], N[(N[(NaChar - NdChar), $MachinePrecision] * N[(N[(NdChar + NaChar), $MachinePrecision] * N[(0.5 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-291], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-270], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / KbT), $MachinePrecision] + N[(Ec * N[(1.0 - N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+27], t$95$0, If[LessEqual[t$95$1, 4e+191], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.99999999999999968e-50

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 40.8

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

   5. Simplified 40.8%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. difference-of-squares N/A

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 23.5

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

   7. Applied egg-rr 23.5%

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 40.8

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

   9. Applied egg-rr 40.8%

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

   if -4.99999999999999968e-50 < (+.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))))) < -1.99999999999999992e-291

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 65.4

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

   5. Simplified 65.4%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 50.0

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

   8. Simplified 50.0%

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

   if -1.99999999999999992e-291 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999998e-270

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 95.5

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

   5. Simplified 95.5%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   8. Simplified 91.1%

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

   9. Taylor expanded in Ec around inf

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

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

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

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

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

       3. mul-1-neg N/A

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

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

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

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

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

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

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

       7. +-lowering-+.f64 93.4

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 + \left(-\frac{EDonor + \color{blue}{\left(Vef + mu\right)}}{Ec}\right)\right)\right)}{KbT}} \]

   11. Simplified 93.4%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \color{blue}{Ec \cdot \left(1 + \left(-\frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}\right)}{KbT}} \]

   if 4.9999999999999998e-270 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1e27 or 4.00000000000000029e191 < (+.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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 65.1

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

   5. Simplified 65.1%

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

   6. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 48.0

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

   8. Simplified 48.0%

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

   if 1e27 < (+.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))))) < 4.00000000000000029e191

   1. Initial program 100.0%

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

   3. Taylor expanded in EAccept around inf

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

   5. Simplified 74.1%

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

   6. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 46.1

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

   8. Simplified 46.1%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{+27}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_2 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-282}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_2 (+ EDonor (+ Vef (- mu Ec)))))
   (if (<= t_1 -5e-50)
     (* (- NaChar NdChar) (* (+ NdChar NaChar) (/ 0.5 (- NaChar NdChar))))
     (if (<= t_1 -1e-210)
       (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
       (if (<= t_1 5e-282)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_2 t_2) KbT)
             (* Ec (- 1.0 (/ (+ EDonor (+ Vef mu)) Ec))))
            KbT)))
         (if (<= t_1 1e+27)
           t_0
           (if (<= t_1 4e+191)
             (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
             t_0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_2 = EDonor + (Vef + (mu - Ec));
	double tmp;
	if (t_1 <= -5e-50) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_1 <= -1e-210) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else if (t_1 <= 5e-282) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_2 * t_2) / KbT), (Ec * (1.0 - ((EDonor + (Vef + mu)) / Ec)))) / KbT));
	} else if (t_1 <= 1e+27) {
		tmp = t_0;
	} else if (t_1 <= 4e+191) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_2 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	tmp = 0.0
	if (t_1 <= -5e-50)
		tmp = Float64(Float64(NaChar - NdChar) * Float64(Float64(NdChar + NaChar) * Float64(0.5 / Float64(NaChar - NdChar))));
	elseif (t_1 <= -1e-210)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (t_1 <= 5e-282)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Float64(Ec * Float64(1.0 - Float64(Float64(EDonor + Float64(Vef + mu)) / Ec)))) / KbT)));
	elseif (t_1 <= 1e+27)
		tmp = t_0;
	elseif (t_1 <= 4e+191)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-50], N[(N[(NaChar - NdChar), $MachinePrecision] * N[(N[(NdChar + NaChar), $MachinePrecision] * N[(0.5 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-210], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-282], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / KbT), $MachinePrecision] + N[(Ec * N[(1.0 - N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+27], t$95$0, If[LessEqual[t$95$1, 4e+191], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.99999999999999968e-50

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 40.8

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

   5. Simplified 40.8%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. difference-of-squares N/A

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 23.5

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

   7. Applied egg-rr 23.5%

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 40.8

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

   9. Applied egg-rr 40.8%

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

   if -4.99999999999999968e-50 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1e-210

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 69.8

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

   5. Simplified 69.8%

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

   6. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 56.1

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

   8. Simplified 56.1%

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

   if -1e-210 < (+.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))))) < 5.0000000000000001e-282

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 93.4

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

   5. Simplified 93.4%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   8. Simplified 86.9%

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

   9. Taylor expanded in Ec around inf

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

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

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

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

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

       3. mul-1-neg N/A

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

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

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

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

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

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

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

       7. +-lowering-+.f64 89.1

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 + \left(-\frac{EDonor + \color{blue}{\left(Vef + mu\right)}}{Ec}\right)\right)\right)}{KbT}} \]

   11. Simplified 89.1%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \color{blue}{Ec \cdot \left(1 + \left(-\frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}\right)}{KbT}} \]

   if 5.0000000000000001e-282 < (+.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))))) < 1e27 or 4.00000000000000029e191 < (+.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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 64.8

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

   5. Simplified 64.8%

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

   6. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 48.2

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

   8. Simplified 48.2%

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

   if 1e27 < (+.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))))) < 4.00000000000000029e191

   1. Initial program 100.0%

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

   3. Taylor expanded in EAccept around inf

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

   5. Simplified 74.1%

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

   6. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 46.1

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

   8. Simplified 46.1%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-282}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{+27}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_1 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;t\_0 \leq 10^{-281}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_1 \cdot t\_1}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_1 (+ EDonor (+ Vef (- mu Ec)))))
   (if (<= t_0 -5e-50)
     (+ (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)) (* NdChar 0.5))
     (if (<= t_0 -2e-291)
       (/ NdChar (+ (exp (/ mu KbT)) 1.0))
       (if (<= t_0 1e-281)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_1 t_1) KbT)
             (* Ec (- 1.0 (/ (+ EDonor (+ Vef mu)) Ec))))
            KbT)))
         (+ (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (* 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 t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_1 = EDonor + (Vef + (mu - Ec));
	double tmp;
	if (t_0 <= -5e-50) {
		tmp = (NaChar / (exp((EAccept / KbT)) + 1.0)) + (NdChar * 0.5);
	} else if (t_0 <= -2e-291) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else if (t_0 <= 1e-281) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_1 * t_1) / KbT), (Ec * (1.0 - ((EDonor + (Vef + mu)) / Ec)))) / KbT));
	} else {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar * 0.5);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_1 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	tmp = 0.0
	if (t_0 <= -5e-50)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)) + Float64(NdChar * 0.5));
	elseif (t_0 <= -2e-291)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	elseif (t_0 <= 1e-281)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_1 * t_1) / KbT), Float64(Ec * Float64(1.0 - Float64(Float64(EDonor + Float64(Vef + mu)) / Ec)))) / KbT)));
	else
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar * 0.5));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-50], N[(N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-291], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-281], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision] + N[(Ec * N[(1.0 - N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $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))))) < -4.99999999999999968e-50

   1. Initial program 99.9%

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

   3. Taylor expanded in EAccept around inf

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

   5. Simplified 81.3%

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

   6. Taylor expanded in KbT around inf

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

      1. *-lowering-*.f64 59.6

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

   8. Simplified 59.6%

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

   if -4.99999999999999968e-50 < (+.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))))) < -1.99999999999999992e-291

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 65.4

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

   5. Simplified 65.4%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 50.0

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

   8. Simplified 50.0%

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

   if -1.99999999999999992e-291 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1e-281

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   8. Simplified 93.2%

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

   9. Taylor expanded in Ec around inf

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

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

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

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

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

       3. mul-1-neg N/A

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

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

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

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

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

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

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

       7. +-lowering-+.f64 95.5

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 + \left(-\frac{EDonor + \color{blue}{\left(Vef + mu\right)}}{Ec}\right)\right)\right)}{KbT}} \]

   11. Simplified 95.5%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \color{blue}{Ec \cdot \left(1 + \left(-\frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}\right)}{KbT}} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 60.0

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

   5. Simplified 60.0%

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

   6. Taylor expanded in EDonor around inf

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

   8. Simplified 48.6%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-281}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_2 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)) (* NdChar 0.5)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_2 (+ EDonor (+ Vef (- mu Ec)))))
   (if (<= t_1 -5e-50)
     t_0
     (if (<= t_1 -2e-291)
       (/ NdChar (+ (exp (/ mu KbT)) 1.0))
       (if (<= t_1 1e-240)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_2 t_2) KbT)
             (* Ec (- 1.0 (/ (+ EDonor (+ Vef mu)) Ec))))
            KbT)))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (exp((EAccept / KbT)) + 1.0)) + (NdChar * 0.5);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_2 = EDonor + (Vef + (mu - Ec));
	double tmp;
	if (t_1 <= -5e-50) {
		tmp = t_0;
	} else if (t_1 <= -2e-291) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else if (t_1 <= 1e-240) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_2 * t_2) / KbT), (Ec * (1.0 - ((EDonor + (Vef + mu)) / Ec)))) / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)) + Float64(NdChar * 0.5))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_2 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	tmp = 0.0
	if (t_1 <= -5e-50)
		tmp = t_0;
	elseif (t_1 <= -2e-291)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	elseif (t_1 <= 1e-240)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Float64(Ec * Float64(1.0 - Float64(Float64(EDonor + Float64(Vef + mu)) / Ec)))) / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-50], t$95$0, If[LessEqual[t$95$1, -2e-291], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-240], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / KbT), $MachinePrecision] + N[(Ec * N[(1.0 - N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.99999999999999968e-50 or 9.9999999999999997e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in EAccept around inf

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

   5. Simplified 76.3%

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

   6. Taylor expanded in KbT around inf

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

      1. *-lowering-*.f64 50.8

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

   8. Simplified 50.8%

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

   if -4.99999999999999968e-50 < (+.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))))) < -1.99999999999999992e-291

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 65.4

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

   5. Simplified 65.4%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 50.0

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

   8. Simplified 50.0%

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

   if -1.99999999999999992e-291 < (+.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))))) < 9.9999999999999997e-241

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 93.7

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

   5. Simplified 93.7%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   8. Simplified 85.3%

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

   9. Taylor expanded in Ec around inf

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

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

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

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

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

       3. mul-1-neg N/A

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

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

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

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

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

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

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

       7. +-lowering-+.f64 87.5

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 + \left(-\frac{EDonor + \color{blue}{\left(Vef + mu\right)}}{Ec}\right)\right)\right)}{KbT}} \]

   11. Simplified 87.5%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_1 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+190}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-233}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;t\_0 \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_1 \cdot t\_1}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_1 (+ EDonor (+ Vef (- mu Ec)))))
   (if (<= t_0 -1e+190)
     (* (- NaChar NdChar) (* (+ NdChar NaChar) (/ 0.5 (- NaChar NdChar))))
     (if (<= t_0 -2e-233)
       (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
       (if (<= t_0 1e-240)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_1 t_1) KbT)
             (* Ec (- 1.0 (/ (+ EDonor (+ Vef mu)) Ec))))
            KbT)))
         (* 0.5 (+ NdChar NaChar)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_1 = EDonor + (Vef + (mu - Ec));
	double tmp;
	if (t_0 <= -1e+190) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_0 <= -2e-233) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (t_0 <= 1e-240) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_1 * t_1) / KbT), (Ec * (1.0 - ((EDonor + (Vef + mu)) / Ec)))) / KbT));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_1 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	tmp = 0.0
	if (t_0 <= -1e+190)
		tmp = Float64(Float64(NaChar - NdChar) * Float64(Float64(NdChar + NaChar) * Float64(0.5 / Float64(NaChar - NdChar))));
	elseif (t_0 <= -2e-233)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (t_0 <= 1e-240)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_1 * t_1) / KbT), Float64(Ec * Float64(1.0 - Float64(Float64(EDonor + Float64(Vef + mu)) / Ec)))) / KbT)));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+190], N[(N[(NaChar - NdChar), $MachinePrecision] * N[(N[(NdChar + NaChar), $MachinePrecision] * N[(0.5 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-233], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-240], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision] + N[(Ec * N[(1.0 - N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $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))))) < -1.0000000000000001e190

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 50.1

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

   5. Simplified 50.1%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. difference-of-squares N/A

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 7.6

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

   7. Applied egg-rr 7.6%

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 50.1

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

   9. Applied egg-rr 50.1%

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

   if -1.0000000000000001e190 < (+.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))))) < -1.99999999999999992e-233

   1. Initial program 99.9%

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

   3. Taylor expanded in EAccept around inf

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

   5. Simplified 73.8%

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

   6. Taylor expanded in NdChar around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 37.2

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

   8. Simplified 37.2%

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

   if -1.99999999999999992e-233 < (+.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))))) < 9.9999999999999997e-241

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 93.9

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

   5. Simplified 93.9%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   8. Simplified 83.6%

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

   9. Taylor expanded in Ec around inf

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

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

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

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

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

       3. mul-1-neg N/A

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

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

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

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

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

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

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

       7. +-lowering-+.f64 85.8

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 + \left(-\frac{EDonor + \color{blue}{\left(Vef + mu\right)}}{Ec}\right)\right)\right)}{KbT}} \]

   11. Simplified 85.8%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \color{blue}{Ec \cdot \left(1 + \left(-\frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}\right)}{KbT}} \]

   if 9.9999999999999997e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 37.6

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

   5. Simplified 37.6%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{+190}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-233}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_1 := t\_0 + \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + t\_0\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-195}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-189}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))
        (t_1 (+ t_0 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          t_0)))
   (if (<= t_2 -5e-195)
     t_1
     (if (<= t_2 5e-189)
       (/ NdChar (+ (exp (/ (+ (+ Vef EDonor) (- mu Ec)) KbT)) 1.0))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0);
	double t_1 = t_0 + (NdChar / (exp((EDonor / KbT)) + 1.0));
	double t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + t_0;
	double tmp;
	if (t_2 <= -5e-195) {
		tmp = t_1;
	} else if (t_2 <= 5e-189) {
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0)
    t_1 = t_0 + (ndchar / (exp((edonor / kbt)) + 1.0d0))
    t_2 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + t_0
    if (t_2 <= (-5d-195)) then
        tmp = t_1
    else if (t_2 <= 5d-189) then
        tmp = ndchar / (exp((((vef + edonor) + (mu - ec)) / kbt)) + 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0);
	double t_1 = t_0 + (NdChar / (Math.exp((EDonor / KbT)) + 1.0));
	double t_2 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + t_0;
	double tmp;
	if (t_2 <= -5e-195) {
		tmp = t_1;
	} else if (t_2 <= 5e-189) {
		tmp = NdChar / (Math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)
	t_1 = t_0 + (NdChar / (math.exp((EDonor / KbT)) + 1.0))
	t_2 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + t_0
	tmp = 0
	if t_2 <= -5e-195:
		tmp = t_1
	elif t_2 <= 5e-189:
		tmp = NdChar / (math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + t_0)
	tmp = 0.0
	if (t_2 <= -5e-195)
		tmp = t_1;
	elseif (t_2 <= 5e-189)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Vef + EDonor) + Float64(mu - Ec)) / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0);
	t_1 = t_0 + (NdChar / (exp((EDonor / KbT)) + 1.0));
	t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + t_0;
	tmp = 0.0;
	if (t_2 <= -5e-195)
		tmp = t_1;
	elseif (t_2 <= 5e-189)
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-195], t$95$1, If[LessEqual[t$95$2, 5e-189], N[(NdChar / N[(N[Exp[N[(N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000009e-195 or 4.9999999999999997e-189 < (+.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

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

   5. Simplified 81.0%

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

   if -5.00000000000000009e-195 < (+.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))))) < 4.9999999999999997e-189

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 88.3

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

   5. Simplified 88.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-189}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (+ Vef (- mu Ec))))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -1e-201)
     (* (- NaChar NdChar) (* (+ NdChar NaChar) (/ 0.5 (- NaChar NdChar))))
     (if (<= t_1 1e-240)
       (/
        NdChar
        (-
         2.0
         (/
          (fma
           -0.5
           (/ (* t_0 t_0) KbT)
           (* Ec (- 1.0 (/ (+ EDonor (+ Vef mu)) Ec))))
          KbT)))
       (* 0.5 (+ NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef + (mu - Ec));
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-201) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_1 <= 1e-240) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), (Ec * (1.0 - ((EDonor + (Vef + mu)) / Ec)))) / KbT));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-201)
		tmp = Float64(Float64(NaChar - NdChar) * Float64(Float64(NdChar + NaChar) * Float64(0.5 / Float64(NaChar - NdChar))));
	elseif (t_1 <= 1e-240)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Ec * Float64(1.0 - Float64(Float64(EDonor + Float64(Vef + mu)) / Ec)))) / KbT)));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-201], N[(N[(NaChar - NdChar), $MachinePrecision] * N[(N[(NdChar + NaChar), $MachinePrecision] * N[(0.5 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-240], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(Ec * N[(1.0 - N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $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))))) < -9.99999999999999946e-202

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 36.8

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

   5. Simplified 36.8%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. difference-of-squares N/A

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 21.9

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

   7. Applied egg-rr 21.9%

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 36.8

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

   9. Applied egg-rr 36.8%

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

   if -9.99999999999999946e-202 < (+.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))))) < 9.9999999999999997e-241

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 90.5

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

   5. Simplified 90.5%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   8. Simplified 77.3%

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

   9. Taylor expanded in Ec around inf

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

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

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

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

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

       3. mul-1-neg N/A

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

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

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

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

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

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

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

       7. +-lowering-+.f64 79.2

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 + \left(-\frac{EDonor + \color{blue}{\left(Vef + mu\right)}}{Ec}\right)\right)\right)}{KbT}} \]

   11. Simplified 79.2%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, \color{blue}{Ec \cdot \left(1 + \left(-\frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}\right)}{KbT}} \]

   if 9.9999999999999997e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 37.6

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

   5. Simplified 37.6%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, Ec \cdot \left(1 - \frac{EDonor + \left(Vef + mu\right)}{Ec}\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef - Ec\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + t\_0}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(Vef + EDonor\right) - \mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (- Vef Ec)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu t_0) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -1e-201)
     (* (- NaChar NdChar) (* (+ NdChar NaChar) (/ 0.5 (- NaChar NdChar))))
     (if (<= t_1 1e-240)
       (/
        NdChar
        (+ 2.0 (/ (- (+ Vef EDonor) (fma -0.5 (/ (* t_0 t_0) KbT) Ec)) KbT)))
       (* 0.5 (+ NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef - Ec);
	double t_1 = (NdChar / (exp(((mu + t_0) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-201) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_1 <= 1e-240) {
		tmp = NdChar / (2.0 + (((Vef + EDonor) - fma(-0.5, ((t_0 * t_0) / KbT), Ec)) / KbT));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef - Ec))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + t_0) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-201)
		tmp = Float64(Float64(NaChar - NdChar) * Float64(Float64(NdChar + NaChar) * Float64(0.5 / Float64(NaChar - NdChar))));
	elseif (t_1 <= 1e-240)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Vef + EDonor) - fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Ec)) / KbT)));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + t$95$0), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-201], N[(N[(NaChar - NdChar), $MachinePrecision] * N[(N[(NdChar + NaChar), $MachinePrecision] * N[(0.5 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-240], N[(NdChar / N[(2.0 + N[(N[(N[(Vef + EDonor), $MachinePrecision] - N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $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))))) < -9.99999999999999946e-202

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 36.8

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

   5. Simplified 36.8%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. difference-of-squares N/A

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 21.9

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

   7. Applied egg-rr 21.9%

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. --lowering--.f64 36.8

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

   9. Applied egg-rr 36.8%

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

   if -9.99999999999999946e-202 < (+.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))))) < 9.9999999999999997e-241

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

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

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

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

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

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

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

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

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 90.5

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

   5. Simplified 90.5%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   8. Simplified 77.3%

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

   9. Taylor expanded in mu around 0

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

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

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

       2. +-commutative N/A

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

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

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

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

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

       5. unpow2 N/A

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

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

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

       7. associate--l+ N/A

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

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

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

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

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

       10. associate--l+ N/A

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

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

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

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

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

       13. +-lowering-+.f64 77.3

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

   11. Simplified 77.3%

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

   if 9.9999999999999997e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 37.6

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

   5. Simplified 37.6%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(Vef + EDonor\right) - \mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef - Ec\right)\right) \cdot \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}, Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(Ec \cdot Ec\right)}{KbT \cdot KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_0 -1e-201)
     (* (- NaChar NdChar) (* (+ NdChar NaChar) (/ 0.5 (- NaChar NdChar))))
     (if (<= t_0 1e-240)
       (/ NdChar (/ (* 0.5 (* Ec Ec)) (* KbT KbT)))
       (* 0.5 (+ NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_0 <= -1e-201) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_0 <= 1e-240) {
		tmp = NdChar / ((0.5 * (Ec * Ec)) / (KbT * KbT));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_0 <= (-1d-201)) then
        tmp = (nachar - ndchar) * ((ndchar + nachar) * (0.5d0 / (nachar - ndchar)))
    else if (t_0 <= 1d-240) then
        tmp = ndchar / ((0.5d0 * (ec * ec)) / (kbt * kbt))
    else
        tmp = 0.5d0 * (ndchar + nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_0 <= -1e-201) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_0 <= 1e-240) {
		tmp = NdChar / ((0.5 * (Ec * Ec)) / (KbT * KbT));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_0 <= -1e-201:
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)))
	elif t_0 <= 1e-240:
		tmp = NdChar / ((0.5 * (Ec * Ec)) / (KbT * KbT))
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_0 <= -1e-201)
		tmp = Float64(Float64(NaChar - NdChar) * Float64(Float64(NdChar + NaChar) * Float64(0.5 / Float64(NaChar - NdChar))));
	elseif (t_0 <= 1e-240)
		tmp = Float64(NdChar / Float64(Float64(0.5 * Float64(Ec * Ec)) / Float64(KbT * KbT)));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_0 <= -1e-201)
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	elseif (t_0 <= 1e-240)
		tmp = NdChar / ((0.5 * (Ec * Ec)) / (KbT * KbT));
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-201], N[(N[(NaChar - NdChar), $MachinePrecision] * N[(N[(NdChar + NaChar), $MachinePrecision] * N[(0.5 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-240], N[(NdChar / N[(N[(0.5 * N[(Ec * Ec), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $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))))) < -9.99999999999999946e-202

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 36.8

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

   5. Simplified 36.8%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. difference-of-squares N/A

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

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

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

      8. +-commutative N/A

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

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

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \color{blue}{\left(NaChar - NdChar\right)}\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      11. --lowering--.f64 21.9

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{\color{blue}{NaChar - NdChar}} \]

   7. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{\frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{NaChar - NdChar}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}} \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(NaChar + NdChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(NaChar - NdChar\right) \cdot \left(NaChar + NdChar\right)\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right)} \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \color{blue}{\left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      8. +-commutative N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \color{blue}{\frac{\frac{1}{2}}{NaChar - NdChar}}\right) \]

      11. --lowering--.f64 36.8

          \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{\color{blue}{NaChar - NdChar}}\right) \]

   9. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)} \]

   if -9.99999999999999946e-202 < (+.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))))) < 9.9999999999999997e-241

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right)} + \left(mu + -1 \cdot Ec\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

      14. --lowering--.f64 90.5

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

   5. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu - Ec\right)}{KbT}}}} \]

   6. Taylor expanded in KbT around -inf

      \[\leadsto \frac{NdChar}{\color{blue}{2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{2 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)}} \]

      2. unsub-neg N/A

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{2 - \color{blue}{\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

   8. Simplified 77.3%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, -\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in Ec around inf

      \[\leadsto \frac{NdChar}{\color{blue}{\frac{1}{2} \cdot \frac{{Ec}^{2}}{{KbT}^{2}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{NdChar}{\color{blue}{\frac{\frac{1}{2} \cdot {Ec}^{2}}{{KbT}^{2}}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{\frac{\frac{1}{2} \cdot {Ec}^{2}}{{KbT}^{2}}}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{NdChar}{\frac{\color{blue}{\frac{1}{2} \cdot {Ec}^{2}}}{{KbT}^{2}}} \]

       4. unpow2 N/A

          \[\leadsto \frac{NdChar}{\frac{\frac{1}{2} \cdot \color{blue}{\left(Ec \cdot Ec\right)}}{{KbT}^{2}}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{NdChar}{\frac{\frac{1}{2} \cdot \color{blue}{\left(Ec \cdot Ec\right)}}{{KbT}^{2}}} \]

       6. unpow2 N/A

          \[\leadsto \frac{NdChar}{\frac{\frac{1}{2} \cdot \left(Ec \cdot Ec\right)}{\color{blue}{KbT \cdot KbT}}} \]

       7. *-lowering-*.f64 51.0

          \[\leadsto \frac{NdChar}{\frac{0.5 \cdot \left(Ec \cdot Ec\right)}{\color{blue}{KbT \cdot KbT}}} \]

   11. Simplified 51.0%

       \[\leadsto \frac{NdChar}{\color{blue}{\frac{0.5 \cdot \left(Ec \cdot Ec\right)}{KbT \cdot KbT}}} \]

   if 9.9999999999999997e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 37.6

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 37.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-240}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(Ec \cdot Ec\right)}{KbT \cdot KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-181}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_0 -1e-201)
     (* (- NaChar NdChar) (* (+ NdChar NaChar) (/ 0.5 (- NaChar NdChar))))
     (if (<= t_0 4e-181)
       (/ NdChar (+ 2.0 (/ (- (+ EDonor (+ Vef mu)) Ec) KbT)))
       (* 0.5 (+ NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_0 <= -1e-201) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_0 <= 4e-181) {
		tmp = NdChar / (2.0 + (((EDonor + (Vef + mu)) - Ec) / KbT));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_0 <= (-1d-201)) then
        tmp = (nachar - ndchar) * ((ndchar + nachar) * (0.5d0 / (nachar - ndchar)))
    else if (t_0 <= 4d-181) then
        tmp = ndchar / (2.0d0 + (((edonor + (vef + mu)) - ec) / kbt))
    else
        tmp = 0.5d0 * (ndchar + nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_0 <= -1e-201) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_0 <= 4e-181) {
		tmp = NdChar / (2.0 + (((EDonor + (Vef + mu)) - Ec) / KbT));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_0 <= -1e-201:
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)))
	elif t_0 <= 4e-181:
		tmp = NdChar / (2.0 + (((EDonor + (Vef + mu)) - Ec) / KbT))
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_0 <= -1e-201)
		tmp = Float64(Float64(NaChar - NdChar) * Float64(Float64(NdChar + NaChar) * Float64(0.5 / Float64(NaChar - NdChar))));
	elseif (t_0 <= 4e-181)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(EDonor + Float64(Vef + mu)) - Ec) / KbT)));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_0 <= -1e-201)
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	elseif (t_0 <= 4e-181)
		tmp = NdChar / (2.0 + (((EDonor + (Vef + mu)) - Ec) / KbT));
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-201], N[(N[(NaChar - NdChar), $MachinePrecision] * N[(N[(NdChar + NaChar), $MachinePrecision] * N[(0.5 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-181], N[(NdChar / N[(2.0 + N[(N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $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))))) < -9.99999999999999946e-202

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 36.8

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 36.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(NaChar + NdChar\right) \cdot \frac{1}{2}} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{NaChar \cdot NaChar - NdChar \cdot NdChar}{NaChar - NdChar}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}}{NaChar - NdChar} \]

      6. difference-of-squares N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \color{blue}{\left(NaChar - NdChar\right)}\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      11. --lowering--.f64 21.9

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{\color{blue}{NaChar - NdChar}} \]

   7. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{\frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{NaChar - NdChar}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}} \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(NaChar + NdChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(NaChar - NdChar\right) \cdot \left(NaChar + NdChar\right)\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right)} \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \color{blue}{\left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      8. +-commutative N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \color{blue}{\frac{\frac{1}{2}}{NaChar - NdChar}}\right) \]

      11. --lowering--.f64 36.8

          \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{\color{blue}{NaChar - NdChar}}\right) \]

   9. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)} \]

   if -9.99999999999999946e-202 < (+.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-181

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right)} + \left(mu + -1 \cdot Ec\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

      14. --lowering--.f64 86.9

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

   5. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu - Ec\right)}{KbT}}}} \]

   6. Taylor expanded in KbT around -inf

      \[\leadsto \frac{NdChar}{\color{blue}{2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{2 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)}} \]

      2. unsub-neg N/A

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{2 - \color{blue}{\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

   8. Simplified 67.2%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, -\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in KbT around inf

      \[\leadsto \frac{NdChar}{2 - \frac{\color{blue}{Ec - \left(EDonor + \left(Vef + mu\right)\right)}}{KbT}} \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\color{blue}{Ec - \left(EDonor + \left(Vef + mu\right)\right)}}{KbT}} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{Ec - \color{blue}{\left(EDonor + \left(Vef + mu\right)\right)}}{KbT}} \]

       3. +-lowering-+.f64 45.3

          \[\leadsto \frac{NdChar}{2 - \frac{Ec - \left(EDonor + \color{blue}{\left(Vef + mu\right)}\right)}{KbT}} \]

   11. Simplified 45.3%

       \[\leadsto \frac{NdChar}{2 - \frac{\color{blue}{Ec - \left(EDonor + \left(Vef + mu\right)\right)}}{KbT}} \]

   if 4.00000000000000019e-181 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 39.3

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 39.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{-181}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-240}:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Ec \cdot Ec}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_0 -1e-201)
     (* (- NaChar NdChar) (* (+ NdChar NaChar) (/ 0.5 (- NaChar NdChar))))
     (if (<= t_0 1e-240)
       (/ (* 2.0 (* NdChar (* KbT KbT))) (* Ec Ec))
       (* 0.5 (+ NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_0 <= -1e-201) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_0 <= 1e-240) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_0 <= (-1d-201)) then
        tmp = (nachar - ndchar) * ((ndchar + nachar) * (0.5d0 / (nachar - ndchar)))
    else if (t_0 <= 1d-240) then
        tmp = (2.0d0 * (ndchar * (kbt * kbt))) / (ec * ec)
    else
        tmp = 0.5d0 * (ndchar + nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_0 <= -1e-201) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_0 <= 1e-240) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_0 <= -1e-201:
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)))
	elif t_0 <= 1e-240:
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec)
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_0 <= -1e-201)
		tmp = Float64(Float64(NaChar - NdChar) * Float64(Float64(NdChar + NaChar) * Float64(0.5 / Float64(NaChar - NdChar))));
	elseif (t_0 <= 1e-240)
		tmp = Float64(Float64(2.0 * Float64(NdChar * Float64(KbT * KbT))) / Float64(Ec * Ec));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_0 <= -1e-201)
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	elseif (t_0 <= 1e-240)
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec);
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-201], N[(N[(NaChar - NdChar), $MachinePrecision] * N[(N[(NdChar + NaChar), $MachinePrecision] * N[(0.5 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-240], N[(N[(2.0 * N[(NdChar * N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Ec * Ec), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $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))))) < -9.99999999999999946e-202

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 36.8

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 36.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(NaChar + NdChar\right) \cdot \frac{1}{2}} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{NaChar \cdot NaChar - NdChar \cdot NdChar}{NaChar - NdChar}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}}{NaChar - NdChar} \]

      6. difference-of-squares N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \color{blue}{\left(NaChar - NdChar\right)}\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      11. --lowering--.f64 21.9

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{\color{blue}{NaChar - NdChar}} \]

   7. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{\frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{NaChar - NdChar}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}} \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(NaChar + NdChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(NaChar - NdChar\right) \cdot \left(NaChar + NdChar\right)\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right)} \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \color{blue}{\left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      8. +-commutative N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \color{blue}{\frac{\frac{1}{2}}{NaChar - NdChar}}\right) \]

      11. --lowering--.f64 36.8

          \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{\color{blue}{NaChar - NdChar}}\right) \]

   9. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)} \]

   if -9.99999999999999946e-202 < (+.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))))) < 9.9999999999999997e-241

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right)} + \left(mu + -1 \cdot Ec\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

      14. --lowering--.f64 90.5

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

   5. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu - Ec\right)}{KbT}}}} \]

   6. Taylor expanded in KbT around -inf

      \[\leadsto \frac{NdChar}{\color{blue}{2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{2 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)}} \]

      2. unsub-neg N/A

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{2 - \color{blue}{\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

   8. Simplified 77.3%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, -\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in Ec around inf

      \[\leadsto \color{blue}{2 \cdot \frac{{KbT}^{2} \cdot NdChar}{{Ec}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}{{Ec}^{2}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}{{Ec}^{2}}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}}{{Ec}^{2}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \color{blue}{\left({KbT}^{2} \cdot NdChar\right)}}{{Ec}^{2}} \]

       5. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(KbT \cdot KbT\right)} \cdot NdChar\right)}{{Ec}^{2}} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(KbT \cdot KbT\right)} \cdot NdChar\right)}{{Ec}^{2}} \]

       7. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\left(KbT \cdot KbT\right) \cdot NdChar\right)}{\color{blue}{Ec \cdot Ec}} \]

       8. *-lowering-*.f64 48.8

          \[\leadsto \frac{2 \cdot \left(\left(KbT \cdot KbT\right) \cdot NdChar\right)}{\color{blue}{Ec \cdot Ec}} \]

   11. Simplified 48.8%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(KbT \cdot KbT\right) \cdot NdChar\right)}{Ec \cdot Ec}} \]

   if 9.9999999999999997e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 37.6

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 37.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-240}:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Ec \cdot Ec}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{EDonor \cdot EDonor}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-291)
     t_0
     (if (<= t_1 0.0)
       (/ (* 2.0 (* NdChar (* KbT KbT))) (* EDonor EDonor))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-291) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (EDonor * EDonor);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-2d-291)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (2.0d0 * (ndchar * (kbt * kbt))) / (edonor * edonor)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-291) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (EDonor * EDonor);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -2e-291:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (EDonor * EDonor)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-291)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(2.0 * Float64(NdChar * Float64(KbT * KbT))) / Float64(EDonor * EDonor));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-291)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (EDonor * EDonor);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-291], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(2.0 * N[(NdChar * N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(EDonor * EDonor), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.99999999999999992e-291 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 35.6

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 35.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -1.99999999999999992e-291 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right)} + \left(mu + -1 \cdot Ec\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

      14. --lowering--.f64 100.0

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu - Ec\right)}{KbT}}}} \]

   6. Taylor expanded in KbT around -inf

      \[\leadsto \frac{NdChar}{\color{blue}{2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{2 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)}} \]

      2. unsub-neg N/A

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{2 - \color{blue}{\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]

   8. Simplified 95.2%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}, -\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in EDonor around inf

      \[\leadsto \color{blue}{2 \cdot \frac{{KbT}^{2} \cdot NdChar}{{EDonor}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}{{EDonor}^{2}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}{{EDonor}^{2}}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}}{{EDonor}^{2}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \color{blue}{\left({KbT}^{2} \cdot NdChar\right)}}{{EDonor}^{2}} \]

       5. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(KbT \cdot KbT\right)} \cdot NdChar\right)}{{EDonor}^{2}} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(KbT \cdot KbT\right)} \cdot NdChar\right)}{{EDonor}^{2}} \]

       7. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\left(KbT \cdot KbT\right) \cdot NdChar\right)}{\color{blue}{EDonor \cdot EDonor}} \]

       8. *-lowering-*.f64 48.2

          \[\leadsto \frac{2 \cdot \left(\left(KbT \cdot KbT\right) \cdot NdChar\right)}{\color{blue}{EDonor \cdot EDonor}} \]

   11. Simplified 48.2%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(KbT \cdot KbT\right) \cdot NdChar\right)}{EDonor \cdot EDonor}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{EDonor \cdot EDonor}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 32.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-248}:\\ \;\;\;\;\frac{0.5 \cdot \left(NaChar \cdot NaChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_0 -1e-210)
     (* (- NaChar NdChar) (* (+ NdChar NaChar) (/ 0.5 (- NaChar NdChar))))
     (if (<= t_0 5e-248)
       (/ (* 0.5 (* NaChar NaChar)) (- NaChar NdChar))
       (* 0.5 (+ NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_0 <= -1e-210) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_0 <= 5e-248) {
		tmp = (0.5 * (NaChar * NaChar)) / (NaChar - NdChar);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_0 <= (-1d-210)) then
        tmp = (nachar - ndchar) * ((ndchar + nachar) * (0.5d0 / (nachar - ndchar)))
    else if (t_0 <= 5d-248) then
        tmp = (0.5d0 * (nachar * nachar)) / (nachar - ndchar)
    else
        tmp = 0.5d0 * (ndchar + nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_0 <= -1e-210) {
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	} else if (t_0 <= 5e-248) {
		tmp = (0.5 * (NaChar * NaChar)) / (NaChar - NdChar);
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_0 <= -1e-210:
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)))
	elif t_0 <= 5e-248:
		tmp = (0.5 * (NaChar * NaChar)) / (NaChar - NdChar)
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_0 <= -1e-210)
		tmp = Float64(Float64(NaChar - NdChar) * Float64(Float64(NdChar + NaChar) * Float64(0.5 / Float64(NaChar - NdChar))));
	elseif (t_0 <= 5e-248)
		tmp = Float64(Float64(0.5 * Float64(NaChar * NaChar)) / Float64(NaChar - NdChar));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_0 <= -1e-210)
		tmp = (NaChar - NdChar) * ((NdChar + NaChar) * (0.5 / (NaChar - NdChar)));
	elseif (t_0 <= 5e-248)
		tmp = (0.5 * (NaChar * NaChar)) / (NaChar - NdChar);
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-210], N[(N[(NaChar - NdChar), $MachinePrecision] * N[(N[(NdChar + NaChar), $MachinePrecision] * N[(0.5 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-248], N[(N[(0.5 * N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision] / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $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))))) < -1e-210

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 36.0

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 36.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(NaChar + NdChar\right) \cdot \frac{1}{2}} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{NaChar \cdot NaChar - NdChar \cdot NdChar}{NaChar - NdChar}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}}{NaChar - NdChar} \]

      6. difference-of-squares N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \color{blue}{\left(NaChar - NdChar\right)}\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      11. --lowering--.f64 21.4

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{\color{blue}{NaChar - NdChar}} \]

   7. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{\frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{NaChar - NdChar}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}} \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(NaChar + NdChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(NaChar - NdChar\right) \cdot \left(NaChar + NdChar\right)\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(NaChar - NdChar\right)} \cdot \left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \color{blue}{\left(\left(NaChar + NdChar\right) \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right)} \]

      8. +-commutative N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \frac{\frac{1}{2}}{NaChar - NdChar}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \color{blue}{\frac{\frac{1}{2}}{NaChar - NdChar}}\right) \]

      11. --lowering--.f64 36.0

          \[\leadsto \left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{\color{blue}{NaChar - NdChar}}\right) \]

   9. Applied egg-rr 36.0%

      \[\leadsto \color{blue}{\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)} \]

   if -1e-210 < (+.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))))) < 5.0000000000000001e-248

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 3.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(NaChar + NdChar\right) \cdot \frac{1}{2}} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{NaChar \cdot NaChar - NdChar \cdot NdChar}{NaChar - NdChar}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}}{NaChar - NdChar} \]

      6. difference-of-squares N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \color{blue}{\left(NaChar - NdChar\right)}\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      11. --lowering--.f64 7.1

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{\color{blue}{NaChar - NdChar}} \]

   7. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{NaChar - NdChar}} \]

   8. Taylor expanded in NdChar around 0

      \[\leadsto \frac{\color{blue}{{NaChar}^{2}} \cdot \frac{1}{2}}{NaChar - NdChar} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      2. *-lowering-*.f64 31.0

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar\right)} \cdot 0.5}{NaChar - NdChar} \]

   10. Simplified 31.0%

       \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar\right)} \cdot 0.5}{NaChar - NdChar} \]

   if 5.0000000000000001e-248 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 37.4

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 37.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\left(NaChar - NdChar\right) \cdot \left(\left(NdChar + NaChar\right) \cdot \frac{0.5}{NaChar - NdChar}\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-248}:\\ \;\;\;\;\frac{0.5 \cdot \left(NaChar \cdot NaChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-240}:\\ \;\;\;\;\frac{0.5 \cdot \left(NaChar \cdot NaChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -1e-208)
     t_0
     (if (<= t_1 1e-240)
       (/ (* 0.5 (* NaChar NaChar)) (- NaChar NdChar))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-208) {
		tmp = t_0;
	} else if (t_1 <= 1e-240) {
		tmp = (0.5 * (NaChar * NaChar)) / (NaChar - NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-1d-208)) then
        tmp = t_0
    else if (t_1 <= 1d-240) then
        tmp = (0.5d0 * (nachar * nachar)) / (nachar - ndchar)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-208) {
		tmp = t_0;
	} else if (t_1 <= 1e-240) {
		tmp = (0.5 * (NaChar * NaChar)) / (NaChar - NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -1e-208:
		tmp = t_0
	elif t_1 <= 1e-240:
		tmp = (0.5 * (NaChar * NaChar)) / (NaChar - NdChar)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-208)
		tmp = t_0;
	elseif (t_1 <= 1e-240)
		tmp = Float64(Float64(0.5 * Float64(NaChar * NaChar)) / Float64(NaChar - NdChar));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -1e-208)
		tmp = t_0;
	elseif (t_1 <= 1e-240)
		tmp = (0.5 * (NaChar * NaChar)) / (NaChar - NdChar);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-208], t$95$0, If[LessEqual[t$95$1, 1e-240], N[(N[(0.5 * N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision] / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.0000000000000001e-208 or 9.9999999999999997e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 37.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 37.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -1.0000000000000001e-208 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999997e-241

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 3.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(NaChar + NdChar\right) \cdot \frac{1}{2}} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{NaChar \cdot NaChar - NdChar \cdot NdChar}{NaChar - NdChar}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}}{NaChar - NdChar} \]

      6. difference-of-squares N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \color{blue}{\left(NaChar - NdChar\right)}\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      11. --lowering--.f64 6.9

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{\color{blue}{NaChar - NdChar}} \]

   7. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{\frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{NaChar - NdChar}} \]

   8. Taylor expanded in NdChar around 0

      \[\leadsto \frac{\color{blue}{{NaChar}^{2}} \cdot \frac{1}{2}}{NaChar - NdChar} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      2. *-lowering-*.f64 29.9

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar\right)} \cdot 0.5}{NaChar - NdChar} \]

   10. Simplified 29.9%

       \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar\right)} \cdot 0.5}{NaChar - NdChar} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-208}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-240}:\\ \;\;\;\;\frac{0.5 \cdot \left(NaChar \cdot NaChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 32.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-240}:\\ \;\;\;\;\frac{-0.5 \cdot \left(NdChar \cdot NdChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -1e-201)
     t_0
     (if (<= t_1 1e-240)
       (/ (* -0.5 (* NdChar NdChar)) (- NaChar NdChar))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-201) {
		tmp = t_0;
	} else if (t_1 <= 1e-240) {
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-1d-201)) then
        tmp = t_0
    else if (t_1 <= 1d-240) then
        tmp = ((-0.5d0) * (ndchar * ndchar)) / (nachar - ndchar)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-201) {
		tmp = t_0;
	} else if (t_1 <= 1e-240) {
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -1e-201:
		tmp = t_0
	elif t_1 <= 1e-240:
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-201)
		tmp = t_0;
	elseif (t_1 <= 1e-240)
		tmp = Float64(Float64(-0.5 * Float64(NdChar * NdChar)) / Float64(NaChar - NdChar));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -1e-201)
		tmp = t_0;
	elseif (t_1 <= 1e-240)
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-201], t$95$0, If[LessEqual[t$95$1, 1e-240], N[(N[(-0.5 * N[(NdChar * NdChar), $MachinePrecision]), $MachinePrecision] / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999946e-202 or 9.9999999999999997e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 37.3

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 37.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -9.99999999999999946e-202 < (+.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))))) < 9.9999999999999997e-241

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 3.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(NaChar + NdChar\right) \cdot \frac{1}{2}} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{NaChar \cdot NaChar - NdChar \cdot NdChar}{NaChar - NdChar}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}{NaChar - NdChar}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(NaChar \cdot NaChar - NdChar \cdot NdChar\right) \cdot \frac{1}{2}}}{NaChar - NdChar} \]

      6. difference-of-squares N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(NaChar + NdChar\right) \cdot \left(NaChar - NdChar\right)\right)} \cdot \frac{1}{2}}{NaChar - NdChar} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(NdChar + NaChar\right)} \cdot \left(NaChar - NdChar\right)\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \color{blue}{\left(NaChar - NdChar\right)}\right) \cdot \frac{1}{2}}{NaChar - NdChar} \]

      11. --lowering--.f64 6.8

          \[\leadsto \frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{\color{blue}{NaChar - NdChar}} \]

   7. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{\frac{\left(\left(NdChar + NaChar\right) \cdot \left(NaChar - NdChar\right)\right) \cdot 0.5}{NaChar - NdChar}} \]

   8. Taylor expanded in NdChar around inf

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot {NdChar}^{2}}}{NaChar - NdChar} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot {NdChar}^{2}}}{NaChar - NdChar} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \color{blue}{\left(NdChar \cdot NdChar\right)}}{NaChar - NdChar} \]

      3. *-lowering-*.f64 23.4

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(NdChar \cdot NdChar\right)}}{NaChar - NdChar} \]

   10. Simplified 23.4%

       \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(NdChar \cdot NdChar\right)}}{NaChar - NdChar} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-201}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 10^{-240}:\\ \;\;\;\;\frac{-0.5 \cdot \left(NdChar \cdot NdChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;0.25 \cdot \frac{mu \cdot NaChar}{KbT}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-291)
     t_0
     (if (<= t_1 0.0) (* 0.25 (/ (* mu NaChar) 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 / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-291) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = 0.25 * ((mu * NaChar) / 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 / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-2d-291)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = 0.25d0 * ((mu * nachar) / 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 / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-291) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = 0.25 * ((mu * NaChar) / 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 / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -2e-291:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = 0.25 * ((mu * NaChar) / 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(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-291)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(0.25 * Float64(Float64(mu * NaChar) / 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 / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-291)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = 0.25 * ((mu * NaChar) / 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[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-291], t$95$0, If[LessEqual[t$95$1, 0.0], N[(0.25 * N[(N[(mu * NaChar), $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))))) < -1.99999999999999992e-291 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 35.6

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 35.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -1.99999999999999992e-291 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]

   5. Simplified 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}, \frac{\left(\left(EDonor + Vef\right) + \left(mu - Ec\right)\right) \cdot NdChar}{KbT}\right), 0.5 \cdot \left(NaChar + NdChar\right)\right)} \]

   6. Taylor expanded in mu around -inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(mu \cdot \left(-1 \cdot \frac{NdChar}{KbT} + \frac{NaChar}{KbT}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot mu\right) \cdot \left(-1 \cdot \frac{NdChar}{KbT} + \frac{NaChar}{KbT}\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot mu\right) \cdot \left(-1 \cdot \frac{NdChar}{KbT} + \frac{NaChar}{KbT}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot mu\right)} \cdot \left(-1 \cdot \frac{NdChar}{KbT} + \frac{NaChar}{KbT}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot mu\right) \cdot \color{blue}{\left(\frac{NaChar}{KbT} + -1 \cdot \frac{NdChar}{KbT}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \left(\frac{1}{4} \cdot mu\right) \cdot \left(\frac{NaChar}{KbT} + \color{blue}{\left(\mathsf{neg}\left(\frac{NdChar}{KbT}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto \left(\frac{1}{4} \cdot mu\right) \cdot \color{blue}{\left(\frac{NaChar}{KbT} - \frac{NdChar}{KbT}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot mu\right) \cdot \color{blue}{\left(\frac{NaChar}{KbT} - \frac{NdChar}{KbT}\right)} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot mu\right) \cdot \left(\color{blue}{\frac{NaChar}{KbT}} - \frac{NdChar}{KbT}\right) \]

      9. /-lowering-/.f64 10.2

         \[\leadsto \left(0.25 \cdot mu\right) \cdot \left(\frac{NaChar}{KbT} - \color{blue}{\frac{NdChar}{KbT}}\right) \]

   8. Simplified 10.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot mu\right) \cdot \left(\frac{NaChar}{KbT} - \frac{NdChar}{KbT}\right)} \]

   9. Taylor expanded in NaChar around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{NaChar \cdot mu}{KbT}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{NaChar \cdot mu}{KbT}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{NaChar \cdot mu}{KbT}} \]

       3. *-commutative N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\color{blue}{mu \cdot NaChar}}{KbT} \]

       4. *-lowering-*.f64 25.9

          \[\leadsto 0.25 \cdot \frac{\color{blue}{mu \cdot NaChar}}{KbT} \]

   11. Simplified 25.9%

       \[\leadsto \color{blue}{0.25 \cdot \frac{mu \cdot NaChar}{KbT}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;0.25 \cdot \frac{mu \cdot NaChar}{KbT}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 69.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 3.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))))
   (if (<= NaChar -1.6e+14)
     t_0
     (if (<= NaChar 3.5e+33)
       (/ NdChar (+ (exp (/ (+ (+ Vef EDonor) (- mu Ec)) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.6e+14) {
		tmp = t_0;
	} else if (NaChar <= 3.5e+33) {
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    if (nachar <= (-1.6d+14)) then
        tmp = t_0
    else if (nachar <= 3.5d+33) then
        tmp = ndchar / (exp((((vef + edonor) + (mu - ec)) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.6e+14) {
		tmp = t_0;
	} else if (NaChar <= 3.5e+33) {
		tmp = NdChar / (Math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -1.6e+14:
		tmp = t_0
	elif NaChar <= 3.5e+33:
		tmp = NdChar / (math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -1.6e+14)
		tmp = t_0;
	elseif (NaChar <= 3.5e+33)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Vef + EDonor) + Float64(mu - Ec)) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -1.6e+14)
		tmp = t_0;
	elseif (NaChar <= 3.5e+33)
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.6e+14], t$95$0, If[LessEqual[NaChar, 3.5e+33], N[(NdChar / N[(N[Exp[N[(N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.6e14 or 3.5000000000000001e33 < NaChar

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept + \left(\left(Ev + Vef\right) - mu\right)}}{KbT}}} \]

      6. +-lowering-+.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. +-lowering-+.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. --lowering--.f64 73.8

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \color{blue}{\left(Vef - mu\right)}\right)}{KbT}}} \]

   5. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}} \]

   if -1.6e14 < NaChar < 3.5000000000000001e33

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right)} + \left(mu + -1 \cdot Ec\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

      14. --lowering--.f64 75.5

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu - Ec\right)}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 3.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 2.5 \cdot 10^{+217}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -5.5e+153)
   (+ (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) (* NaChar 0.5))
   (if (<= KbT 2.5e+217)
     (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
     (+ (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)) (* NdChar 0.5)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -5.5e+153) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar * 0.5);
	} else if (KbT <= 2.5e+217) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = (NaChar / (exp((EAccept / KbT)) + 1.0)) + (NdChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-5.5d+153)) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar * 0.5d0)
    else if (kbt <= 2.5d+217) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else
        tmp = (nachar / (exp((eaccept / kbt)) + 1.0d0)) + (ndchar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -5.5e+153) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar * 0.5);
	} else if (KbT <= 2.5e+217) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = (NaChar / (Math.exp((EAccept / KbT)) + 1.0)) + (NdChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -5.5e+153:
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar * 0.5)
	elif KbT <= 2.5e+217:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	else:
		tmp = (NaChar / (math.exp((EAccept / KbT)) + 1.0)) + (NdChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -5.5e+153)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar * 0.5));
	elseif (KbT <= 2.5e+217)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)) + Float64(NdChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -5.5e+153)
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar * 0.5);
	elseif (KbT <= 2.5e+217)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	else
		tmp = (NaChar / (exp((EAccept / KbT)) + 1.0)) + (NdChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -5.5e+153], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.5e+217], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -5.5000000000000003e153

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}{KbT}}} + \color{blue}{\frac{1}{2} \cdot NaChar} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}{KbT}}} + \color{blue}{NaChar \cdot \frac{1}{2}} \]

      2. *-lowering-*.f64 82.2

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \color{blue}{NaChar \cdot 0.5} \]

   5. Simplified 82.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \color{blue}{NaChar \cdot 0.5} \]

   6. Taylor expanded in EDonor around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor}}{KbT}}} + NaChar \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Simplified 79.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor}}{KbT}}} + NaChar \cdot 0.5 \]

   if -5.5000000000000003e153 < KbT < 2.50000000000000021e217

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept + \left(\left(Ev + Vef\right) - mu\right)}}{KbT}}} \]

      6. +-lowering-+.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. +-lowering-+.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. --lowering--.f64 62.5

          \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \color{blue}{\left(Vef - mu\right)}\right)}{KbT}}} \]

   5. Simplified 62.5%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}}}} \]

   if 2.50000000000000021e217 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in EAccept around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept}}{KbT}}} \]
   4. Step-by-step derivation

   5. Simplified 97.3%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept}}{KbT}}} \]

   6. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 87.1

         \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]

   8. Simplified 87.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 2.5 \cdot 10^{+217}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 40.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.28 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, Ev \cdot \frac{NaChar}{KbT}, t\_0\right)\\ \mathbf{elif}\;KbT \leq -1750:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 2.55 \cdot 10^{+101}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.28e+154)
     (fma -0.25 (* Ev (/ NaChar KbT)) t_0)
     (if (<= KbT -1750.0)
       (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
       (if (<= KbT 2.55e+101) (/ NdChar (+ (exp (/ Ec (- KbT))) 1.0)) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.28e+154) {
		tmp = fma(-0.25, (Ev * (NaChar / KbT)), t_0);
	} else if (KbT <= -1750.0) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (KbT <= 2.55e+101) {
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.28e+154)
		tmp = fma(-0.25, Float64(Ev * Float64(NaChar / KbT)), t_0);
	elseif (KbT <= -1750.0)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (KbT <= 2.55e+101)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.28e+154], N[(-0.25 * N[(Ev * N[(NaChar / KbT), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[KbT, -1750.0], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.55e+101], N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -1.2800000000000001e154

   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 58.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}, \frac{\left(\left(EDonor + Vef\right) + \left(mu - Ec\right)\right) \cdot NdChar}{KbT}\right), 0.5 \cdot \left(NaChar + NdChar\right)\right)} \]

   6. Taylor expanded in Ev around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\frac{Ev \cdot NaChar}{KbT}}, \frac{1}{2} \cdot \left(NaChar + NdChar\right)\right) \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{Ev \cdot \frac{NaChar}{KbT}}, \frac{1}{2} \cdot \left(NaChar + NdChar\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{Ev \cdot \frac{NaChar}{KbT}}, \frac{1}{2} \cdot \left(NaChar + NdChar\right)\right) \]

      3. /-lowering-/.f64 70.2

         \[\leadsto \mathsf{fma}\left(-0.25, Ev \cdot \color{blue}{\frac{NaChar}{KbT}}, 0.5 \cdot \left(NaChar + NdChar\right)\right) \]

   8. Simplified 70.2%

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{Ev \cdot \frac{NaChar}{KbT}}, 0.5 \cdot \left(NaChar + NdChar\right)\right) \]

   if -1.2800000000000001e154 < KbT < -1750

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in EAccept around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept}}{KbT}}} \]
   4. Step-by-step derivation

   5. Simplified 64.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept}}{KbT}}} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{EAccept}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{EAccept}{KbT}}}} \]

      4. /-lowering-/.f64 59.2

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   8. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}} \]

   if -1750 < KbT < 2.54999999999999997e101

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right)} + \left(mu + -1 \cdot Ec\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

      14. --lowering--.f64 66.0

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu - Ec\right)}{KbT}}}} \]

   6. Taylor expanded in Ec around inf

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{-1 \cdot \frac{Ec}{KbT}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{-1 \cdot Ec}{KbT}}}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(Ec\right)}}{KbT}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(Ec\right)}{KbT}}}} \]

      4. neg-lowering-neg.f64 41.8

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{-Ec}}{KbT}}} \]

   8. Simplified 41.8%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{-Ec}{KbT}}}} \]

   if 2.54999999999999997e101 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 64.2

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 64.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.28 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, Ev \cdot \frac{NaChar}{KbT}, 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \mathbf{elif}\;KbT \leq -1750:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 2.55 \cdot 10^{+101}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 40.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -4.3 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, Ev \cdot \frac{NaChar}{KbT}, t\_0\right)\\ \mathbf{elif}\;KbT \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 1.35 \cdot 10^{+117}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -4.3e+153)
     (fma -0.25 (* Ev (/ NaChar KbT)) t_0)
     (if (<= KbT -1.55e-27)
       (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
       (if (<= KbT 1.35e+117) (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -4.3e+153) {
		tmp = fma(-0.25, (Ev * (NaChar / KbT)), t_0);
	} else if (KbT <= -1.55e-27) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (KbT <= 1.35e+117) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -4.3e+153)
		tmp = fma(-0.25, Float64(Ev * Float64(NaChar / KbT)), t_0);
	elseif (KbT <= -1.55e-27)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (KbT <= 1.35e+117)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -4.3e+153], N[(-0.25 * N[(Ev * N[(NaChar / KbT), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[KbT, -1.55e-27], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.35e+117], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -4.2999999999999998e153

   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 58.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}, \frac{\left(\left(EDonor + Vef\right) + \left(mu - Ec\right)\right) \cdot NdChar}{KbT}\right), 0.5 \cdot \left(NaChar + NdChar\right)\right)} \]

   6. Taylor expanded in Ev around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\frac{Ev \cdot NaChar}{KbT}}, \frac{1}{2} \cdot \left(NaChar + NdChar\right)\right) \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{Ev \cdot \frac{NaChar}{KbT}}, \frac{1}{2} \cdot \left(NaChar + NdChar\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{Ev \cdot \frac{NaChar}{KbT}}, \frac{1}{2} \cdot \left(NaChar + NdChar\right)\right) \]

      3. /-lowering-/.f64 70.2

         \[\leadsto \mathsf{fma}\left(-0.25, Ev \cdot \color{blue}{\frac{NaChar}{KbT}}, 0.5 \cdot \left(NaChar + NdChar\right)\right) \]

   8. Simplified 70.2%

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{Ev \cdot \frac{NaChar}{KbT}}, 0.5 \cdot \left(NaChar + NdChar\right)\right) \]

   if -4.2999999999999998e153 < KbT < -1.5499999999999999e-27

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in EAccept around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept}}{KbT}}} \]
   4. Step-by-step derivation

   5. Simplified 64.8%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{EAccept}}{KbT}}} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{1 + e^{\frac{EAccept}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NaChar}{1 + \color{blue}{e^{\frac{EAccept}{KbT}}}} \]

      4. /-lowering-/.f64 52.5

         \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   8. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}} \]

   if -1.5499999999999999e-27 < KbT < 1.3500000000000001e117

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

      5. associate--l+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor + \left(\left(Vef + mu\right) - Ec\right)}}{KbT}}} \]

      6. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(Vef + \left(mu + \color{blue}{-1 \cdot Ec}\right)\right)}{KbT}}} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right) + \left(mu + -1 \cdot Ec\right)}}{KbT}}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(EDonor + Vef\right)} + \left(mu + -1 \cdot Ec\right)}{KbT}}} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      13. sub-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

      14. --lowering--.f64 65.3

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \color{blue}{\left(mu - Ec\right)}}{KbT}}} \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) + \left(mu - Ec\right)}{KbT}}}} \]

   6. Taylor expanded in EDonor around inf

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 39.4

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]

   8. Simplified 39.4%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]

   if 1.3500000000000001e117 < KbT

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 64.0

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 64.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.3 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, Ev \cdot \frac{NaChar}{KbT}, 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \mathbf{elif}\;KbT \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 1.35 \cdot 10^{+117}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 22.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -13:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq 8 \cdot 10^{-65}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -13.0)
   (* NaChar 0.5)
   (if (<= NaChar 8e-65) (* 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 <= -13.0) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 8e-65) {
		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 <= (-13.0d0)) then
        tmp = nachar * 0.5d0
    else if (nachar <= 8d-65) 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 <= -13.0) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 8e-65) {
		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 <= -13.0:
		tmp = NaChar * 0.5
	elif NaChar <= 8e-65:
		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 <= -13.0)
		tmp = Float64(NaChar * 0.5);
	elseif (NaChar <= 8e-65)
		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 <= -13.0)
		tmp = NaChar * 0.5;
	elseif (NaChar <= 8e-65)
		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, -13.0], N[(NaChar * 0.5), $MachinePrecision], If[LessEqual[NaChar, 8e-65], N[(NdChar * 0.5), $MachinePrecision], N[(NaChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -13 or 7.99999999999999939e-65 < 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. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 31.4

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 31.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   6. Taylor expanded in NaChar around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{NaChar \cdot \frac{1}{2}} \]

      2. *-lowering-*.f64 26.8

         \[\leadsto \color{blue}{NaChar \cdot 0.5} \]

   8. Simplified 26.8%

      \[\leadsto \color{blue}{NaChar \cdot 0.5} \]

   if -13 < NaChar < 7.99999999999999939e-65

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

      3. +-lowering-+.f64 29.8

         \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   5. Simplified 29.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   6. Taylor expanded in NaChar around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{NdChar \cdot \frac{1}{2}} \]

      2. *-lowering-*.f64 28.2

         \[\leadsto \color{blue}{NdChar \cdot 0.5} \]

   8. Simplified 28.2%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 27.3% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(NdChar + NaChar\right) \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* 0.5 (+ NdChar NaChar)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return 0.5 * (NdChar + NaChar);
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = 0.5d0 * (ndchar + nachar)
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return 0.5 * (NdChar + NaChar);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return 0.5 * (NdChar + NaChar)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(0.5 * Float64(NdChar + NaChar))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.5 * (NdChar + NaChar);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
2. Add Preprocessing

3. Taylor expanded in KbT around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   3. +-lowering-+.f64 30.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

5. Simplified 30.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

6. Final simplification 30.7%

   \[\leadsto 0.5 \cdot \left(NdChar + NaChar\right) \]
7. Add Preprocessing

Alternative 24: 18.2% accurate, 46.0× speedup

Math

\[\begin{array}{l} \\ NaChar \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* NaChar 0.5))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NaChar * 0.5;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = nachar * 0.5d0
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NaChar * 0.5;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return NaChar * 0.5

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(NaChar * 0.5)
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = NaChar * 0.5;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(NaChar * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
2. Add Preprocessing

3. Taylor expanded in KbT around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NaChar + NdChar\right)} \]

   3. +-lowering-+.f64 30.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

5. Simplified 30.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

6. Taylor expanded in NaChar around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{NaChar \cdot \frac{1}{2}} \]

   2. *-lowering-*.f64 21.0

      \[\leadsto \color{blue}{NaChar \cdot 0.5} \]

8. Simplified 21.0%

   \[\leadsto \color{blue}{NaChar \cdot 0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(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))))))