Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 18.5s

Alternatives: 16

Speedup: 1.0×

• could not determine a ground truth

  1. branch26916 = 1.1694924408008172e-165
  2. NdChar = 4.284838299556698e-165
  3. Ec = 1.7778814818606362e+279
  4. Vef = -4.621473145401323e+107
  5. EDonor = 1.505930502259667e-170
  6. mu = -1.3476072636446465e-188
  7. KbT = 1.2251210019283793e+144
  8. NaChar = -2.217827536844626e+38
  9. Ev = -6.134162085274895e+257
  10. EAccept = -3.8112503248266326e+202

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

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

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NdChar (+ (exp (/ (- mu (- (- Ec Vef) EDonor)) 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 - ((Ec - Vef) - EDonor)) / 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 - ((ec - vef) - edonor)) / 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 - ((Ec - Vef) - EDonor)) / 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 - ((Ec - Vef) - EDonor)) / 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(Float64(Ec - Vef) - EDonor)) / 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 - ((Ec - Vef) - EDonor)) / 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[(N[(Ec - Vef), $MachinePrecision] - EDonor), $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%

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

Alternative 2: 45.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ t_2 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ t_3 := Ec - \left(Vef + mu\right)\\ t_4 := EDonor - t\_3\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-274}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_4 \cdot t\_4}{KbT}, t\_3 - EDonor\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, Vef \cdot \frac{NdChar}{KbT}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0))))
        (t_2 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)))
        (t_3 (- Ec (+ Vef mu)))
        (t_4 (- EDonor t_3)))
   (if (<= t_1 -2e-40)
     t_0
     (if (<= t_1 -2e-274)
       t_2
       (if (<= t_1 0.0)
         (/
          NdChar
          (- 2.0 (/ (fma -0.5 (/ (* t_4 t_4) KbT) (- t_3 EDonor)) KbT)))
         (if (<= t_1 2e+183) t_2 (fma -0.25 (* Vef (/ NdChar 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 - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double t_2 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double t_3 = Ec - (Vef + mu);
	double t_4 = EDonor - t_3;
	double tmp;
	if (t_1 <= -2e-40) {
		tmp = t_0;
	} else if (t_1 <= -2e-274) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_4 * t_4) / KbT), (t_3 - EDonor)) / KbT));
	} else if (t_1 <= 2e+183) {
		tmp = t_2;
	} else {
		tmp = fma(-0.25, (Vef * (NdChar / KbT)), 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(Float64(Ec - Vef) - EDonor)) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	t_2 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	t_3 = Float64(Ec - Float64(Vef + mu))
	t_4 = Float64(EDonor - t_3)
	tmp = 0.0
	if (t_1 <= -2e-40)
		tmp = t_0;
	elseif (t_1 <= -2e-274)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_4 * t_4) / KbT), Float64(t_3 - EDonor)) / KbT)));
	elseif (t_1 <= 2e+183)
		tmp = t_2;
	else
		tmp = fma(-0.25, Float64(Vef * Float64(NdChar / KbT)), t_0);
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $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[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(EDonor - t$95$3), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-40], t$95$0, If[LessEqual[t$95$1, -2e-274], t$95$2, If[LessEqual[t$95$1, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$4 * t$95$4), $MachinePrecision] / KbT), $MachinePrecision] + N[(t$95$3 - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+183], t$95$2, N[(-0.25 * N[(Vef * N[(NdChar / KbT), $MachinePrecision]), $MachinePrecision] + t$95$0), $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))))) < -1.9999999999999999e-40

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 31.0

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

   5. Applied rewrites 31.0%

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

   if -1.9999999999999999e-40 < (+.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))))) < -1.99999999999999993e-274 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))))) < 1.99999999999999989e183

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 40.6%

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

   if -1.99999999999999993e-274 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 92.2%

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

   if 1.99999999999999989e183 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around -inf

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

   5. Applied rewrites 21.0%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 37.3%

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

   9. Taylor expanded in NaChar around 0

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

   11. Applied rewrites 42.5%

       \[\leadsto \mathsf{fma}\left(-0.25, Vef \cdot \frac{NdChar}{KbT}, 0.5 \cdot \left(NaChar + NdChar\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-274}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor - \left(Ec - \left(Vef + mu\right)\right)\right) \cdot \left(EDonor - \left(Ec - \left(Vef + mu\right)\right)\right)}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, Vef \cdot \frac{NdChar}{KbT}, 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := Ec - \left(Vef + mu\right)\\ t_1 := EDonor - t\_0\\ t_2 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_3 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-267}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_1 \cdot t\_1}{KbT}, t\_0 - EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- Ec (+ Vef mu)))
        (t_1 (- EDonor t_0))
        (t_2 (* 0.5 (+ NdChar NaChar)))
        (t_3
         (+
          (/ NdChar (+ (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_3 -5e-267)
     t_2
     (if (<= t_3 2e-286)
       (/ NdChar (- 2.0 (/ (fma -0.5 (/ (* t_1 t_1) KbT) (- t_0 EDonor)) KbT)))
       t_2))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Ec - (Vef + mu);
	double t_1 = EDonor - t_0;
	double t_2 = 0.5 * (NdChar + NaChar);
	double t_3 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_3 <= -5e-267) {
		tmp = t_2;
	} else if (t_3 <= 2e-286) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_1 * t_1) / KbT), (t_0 - EDonor)) / KbT));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Ec - Float64(Vef + mu))
	t_1 = Float64(EDonor - t_0)
	t_2 = Float64(0.5 * Float64(NdChar + NaChar))
	t_3 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_3 <= -5e-267)
		tmp = t_2;
	elseif (t_3 <= 2e-286)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_1 * t_1) / KbT), Float64(t_0 - EDonor)) / KbT)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(EDonor - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $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$3, -5e-267], t$95$2, If[LessEqual[t$95$3, 2e-286], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision] + N[(t$95$0 - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.9999999999999999e-267 or 2.0000000000000001e-286 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 32.1

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

   5. Applied rewrites 32.1%

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

   if -4.9999999999999999e-267 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.0000000000000001e-286

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 84.4%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor - \left(Ec - \left(Vef + mu\right)\right)\right) \cdot \left(EDonor - \left(Ec - \left(Vef + mu\right)\right)\right)}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 35.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-267}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\frac{NdChar}{2 - \left(\left(\frac{Ec - mu}{KbT} - \frac{Vef}{KbT}\right) - \frac{EDonor}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -5e-267)
     t_0
     (if (<= t_1 5e-158)
       (/ NdChar (- 2.0 (- (- (/ (- Ec mu) KbT) (/ Vef KbT)) (/ EDonor KbT))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-267) {
		tmp = t_0;
	} else if (t_1 <= 5e-158) {
		tmp = NdChar / (2.0 - ((((Ec - mu) / KbT) - (Vef / KbT)) - (EDonor / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (exp(((mu - ((ec - vef) - edonor)) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-5d-267)) then
        tmp = t_0
    else if (t_1 <= 5d-158) then
        tmp = ndchar / (2.0d0 - ((((ec - mu) / kbt) - (vef / kbt)) - (edonor / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-267) {
		tmp = t_0;
	} else if (t_1 <= 5e-158) {
		tmp = NdChar / (2.0 - ((((Ec - mu) / KbT) - (Vef / KbT)) - (EDonor / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -5e-267:
		tmp = t_0
	elif t_1 <= 5e-158:
		tmp = NdChar / (2.0 - ((((Ec - mu) / KbT) - (Vef / KbT)) - (EDonor / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e-267)
		tmp = t_0;
	elseif (t_1 <= 5e-158)
		tmp = Float64(NdChar / Float64(2.0 - Float64(Float64(Float64(Float64(Ec - mu) / KbT) - Float64(Vef / KbT)) - Float64(EDonor / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -5e-267)
		tmp = t_0;
	elseif (t_1 <= 5e-158)
		tmp = NdChar / (2.0 - ((((Ec - mu) / KbT) - (Vef / KbT)) - (EDonor / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-267], t$95$0, If[LessEqual[t$95$1, 5e-158], N[(NdChar / N[(2.0 - N[(N[(N[(N[(Ec - mu), $MachinePrecision] / KbT), $MachinePrecision] - N[(Vef / KbT), $MachinePrecision]), $MachinePrecision] - N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.9999999999999999e-267 or 4.99999999999999972e-158 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 32.4

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

   5. Applied rewrites 32.4%

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

   if -4.9999999999999999e-267 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999972e-158

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in KbT around inf

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

   11. Applied rewrites 48.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\frac{NdChar}{2 - \left(\left(\frac{Ec - mu}{KbT} - \frac{Vef}{KbT}\right) - \frac{EDonor}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-267}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-286}:\\ \;\;\;\;0.5 \cdot \left(\left(NaChar \cdot NaChar\right) \cdot \frac{1}{NaChar - NdChar}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ (+ Vef Ev) EAccept) mu) KbT)) 1.0)))))
   (if (<= t_1 -5e-267)
     t_0
     (if (<= t_1 2e-286)
       (* 0.5 (* (* NaChar NaChar) (/ 1.0 (- 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 - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-267) {
		tmp = t_0;
	} else if (t_1 <= 2e-286) {
		tmp = 0.5 * ((NaChar * NaChar) * (1.0 / (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 - ((ec - vef) - edonor)) / kbt)) + 1.0d0)) + (nachar / (exp(((((vef + ev) + eaccept) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-5d-267)) then
        tmp = t_0
    else if (t_1 <= 2d-286) then
        tmp = 0.5d0 * ((nachar * nachar) * (1.0d0 / (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 - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (Math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-267) {
		tmp = t_0;
	} else if (t_1 <= 2e-286) {
		tmp = 0.5 * ((NaChar * NaChar) * (1.0 / (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 - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (math.exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -5e-267:
		tmp = t_0
	elif t_1 <= 2e-286:
		tmp = 0.5 * ((NaChar * NaChar) * (1.0 / (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(Float64(Ec - Vef) - EDonor)) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Vef + Ev) + EAccept) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e-267)
		tmp = t_0;
	elseif (t_1 <= 2e-286)
		tmp = Float64(0.5 * Float64(Float64(NaChar * NaChar) * Float64(1.0 / 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 - ((Ec - Vef) - EDonor)) / KbT)) + 1.0)) + (NaChar / (exp(((((Vef + Ev) + EAccept) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -5e-267)
		tmp = t_0;
	elseif (t_1 <= 2e-286)
		tmp = 0.5 * ((NaChar * NaChar) * (1.0 / (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[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-267], t$95$0, If[LessEqual[t$95$1, 2e-286], N[(0.5 * N[(N[(NaChar * NaChar), $MachinePrecision] * N[(1.0 / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.9999999999999999e-267 or 2.0000000000000001e-286 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 32.1

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

   5. Applied rewrites 32.1%

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

   if -4.9999999999999999e-267 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.0000000000000001e-286

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

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

   7. Applied rewrites 5.9%

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

   8. Taylor expanded in NdChar around 0

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

   10. Applied rewrites 37.7%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-286}:\\ \;\;\;\;0.5 \cdot \left(\left(NaChar \cdot NaChar\right) \cdot \frac{1}{NaChar - NdChar}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{if}\;mu \leq -1.18 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq 5.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) + EAccept}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(Vef - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar (+ (exp (/ (- (+ Ev EAccept) mu) KbT)) 1.0))
          (/ NdChar (+ (exp (/ (+ EDonor (- mu Ec)) KbT)) 1.0)))))
   (if (<= mu -1.18e+116)
     t_0
     (if (<= mu 5.4e+122)
       (+
        (/ NaChar (+ (exp (/ (+ (+ Vef Ev) EAccept) KbT)) 1.0))
        (/ NdChar (+ (exp (/ (+ EDonor (- Vef Ec)) KbT)) 1.0)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (exp((((Ev + EAccept) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (mu - Ec)) / KbT)) + 1.0));
	double tmp;
	if (mu <= -1.18e+116) {
		tmp = t_0;
	} else if (mu <= 5.4e+122) {
		tmp = (NaChar / (exp((((Vef + Ev) + EAccept) / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (Vef - Ec)) / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (nachar / (exp((((ev + eaccept) - mu) / kbt)) + 1.0d0)) + (ndchar / (exp(((edonor + (mu - ec)) / kbt)) + 1.0d0))
    if (mu <= (-1.18d+116)) then
        tmp = t_0
    else if (mu <= 5.4d+122) then
        tmp = (nachar / (exp((((vef + ev) + eaccept) / kbt)) + 1.0d0)) + (ndchar / (exp(((edonor + (vef - ec)) / kbt)) + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (Math.exp((((Ev + EAccept) - mu) / KbT)) + 1.0)) + (NdChar / (Math.exp(((EDonor + (mu - Ec)) / KbT)) + 1.0));
	double tmp;
	if (mu <= -1.18e+116) {
		tmp = t_0;
	} else if (mu <= 5.4e+122) {
		tmp = (NaChar / (Math.exp((((Vef + Ev) + EAccept) / KbT)) + 1.0)) + (NdChar / (Math.exp(((EDonor + (Vef - Ec)) / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (math.exp((((Ev + EAccept) - mu) / KbT)) + 1.0)) + (NdChar / (math.exp(((EDonor + (mu - Ec)) / KbT)) + 1.0))
	tmp = 0
	if mu <= -1.18e+116:
		tmp = t_0
	elif mu <= 5.4e+122:
		tmp = (NaChar / (math.exp((((Vef + Ev) + EAccept) / KbT)) + 1.0)) + (NdChar / (math.exp(((EDonor + (Vef - 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(Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) - mu) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(mu - Ec)) / KbT)) + 1.0)))
	tmp = 0.0
	if (mu <= -1.18e+116)
		tmp = t_0;
	elseif (mu <= 5.4e+122)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Ev) + EAccept) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef - Ec)) / KbT)) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (exp((((Ev + EAccept) - mu) / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (mu - Ec)) / KbT)) + 1.0));
	tmp = 0.0;
	if (mu <= -1.18e+116)
		tmp = t_0;
	elseif (mu <= 5.4e+122)
		tmp = (NaChar / (exp((((Vef + Ev) + EAccept) / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (Vef - 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[(N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(mu - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -1.18e+116], t$95$0, If[LessEqual[mu, 5.4e+122], N[(N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if mu < -1.1799999999999999e116 or 5.3999999999999997e122 < mu

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 91.1%

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

   if -1.1799999999999999e116 < mu < 5.3999999999999997e122

   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 mu around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate--l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.18 \cdot 10^{+116}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 5.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) + EAccept}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(Vef - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) - mu}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(mu - Ec\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{if}\;mu \leq -2.65 \cdot 10^{+162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq 9.6 \cdot 10^{+169}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) + EAccept}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(Vef - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)) 1.0))))
   (if (<= mu -2.65e+162)
     t_0
     (if (<= mu 9.6e+169)
       (+
        (/ NaChar (+ (exp (/ (+ (+ Vef Ev) EAccept) KbT)) 1.0))
        (/ NdChar (+ (exp (/ (+ EDonor (- Vef Ec)) KbT)) 1.0)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (mu <= -2.65e+162) {
		tmp = t_0;
	} else if (mu <= 9.6e+169) {
		tmp = (NaChar / (exp((((Vef + Ev) + EAccept) / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (Vef - Ec)) / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp((((ev + eaccept) + (vef - mu)) / kbt)) + 1.0d0)
    if (mu <= (-2.65d+162)) then
        tmp = t_0
    else if (mu <= 9.6d+169) then
        tmp = (nachar / (exp((((vef + ev) + eaccept) / kbt)) + 1.0d0)) + (ndchar / (exp(((edonor + (vef - ec)) / kbt)) + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (mu <= -2.65e+162) {
		tmp = t_0;
	} else if (mu <= 9.6e+169) {
		tmp = (NaChar / (Math.exp((((Vef + Ev) + EAccept) / KbT)) + 1.0)) + (NdChar / (Math.exp(((EDonor + (Vef - Ec)) / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0)
	tmp = 0
	if mu <= -2.65e+162:
		tmp = t_0
	elif mu <= 9.6e+169:
		tmp = (NaChar / (math.exp((((Vef + Ev) + EAccept) / KbT)) + 1.0)) + (NdChar / (math.exp(((EDonor + (Vef - Ec)) / KbT)) + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)) + 1.0))
	tmp = 0.0
	if (mu <= -2.65e+162)
		tmp = t_0;
	elseif (mu <= 9.6e+169)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Ev) + EAccept) / KbT)) + 1.0)) + Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef - Ec)) / KbT)) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	tmp = 0.0;
	if (mu <= -2.65e+162)
		tmp = t_0;
	elseif (mu <= 9.6e+169)
		tmp = (NaChar / (exp((((Vef + Ev) + EAccept) / KbT)) + 1.0)) + (NdChar / (exp(((EDonor + (Vef - Ec)) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -2.65e+162], t$95$0, If[LessEqual[mu, 9.6e+169], N[(N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] + EAccept), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if mu < -2.6500000000000001e162 or 9.5999999999999994e169 < mu

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -2.6500000000000001e162 < mu < 9.5999999999999994e169

   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 mu around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate--l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2.65 \cdot 10^{+162}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 9.6 \cdot 10^{+169}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) + EAccept}{KbT}} + 1} + \frac{NdChar}{e^{\frac{EDonor + \left(Vef - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -2.4 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 1.7 \cdot 10^{-252}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 3.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)) 1.0))))
   (if (<= NaChar -2.4e-214)
     t_0
     (if (<= NaChar 1.7e-252)
       (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
       (if (<= NaChar 3.2e-166)
         (/ 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 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -2.4e-214) {
		tmp = t_0;
	} else if (NaChar <= 1.7e-252) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else if (NaChar <= 3.2e-166) {
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp((((ev + eaccept) + (vef - mu)) / kbt)) + 1.0d0)
    if (nachar <= (-2.4d-214)) then
        tmp = t_0
    else if (nachar <= 1.7d-252) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    else if (nachar <= 3.2d-166) then
        tmp = ndchar / (exp((ec / -kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -2.4e-214) {
		tmp = t_0;
	} else if (NaChar <= 1.7e-252) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else if (NaChar <= 3.2e-166) {
		tmp = NdChar / (Math.exp((Ec / -KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -2.4e-214:
		tmp = t_0
	elif NaChar <= 1.7e-252:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	elif NaChar <= 3.2e-166:
		tmp = NdChar / (math.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(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -2.4e-214)
		tmp = t_0;
	elseif (NaChar <= 1.7e-252)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	elseif (NaChar <= 3.2e-166)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -2.4e-214)
		tmp = t_0;
	elseif (NaChar <= 1.7e-252)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	elseif (NaChar <= 3.2e-166)
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -2.4e-214], t$95$0, If[LessEqual[NaChar, 1.7e-252], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 3.2e-166], N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -2.4000000000000002e-214 or 3.20000000000000001e-166 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if -2.4000000000000002e-214 < NaChar < 1.7e-252

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 62.1%

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

   if 1.7e-252 < NaChar < 3.20000000000000001e-166

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in Ec around inf

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

   8. Applied rewrites 71.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.4 \cdot 10^{-214}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 1.7 \cdot 10^{-252}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 3.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -3.55 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq -1.52 \cdot 10^{-226}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;Vef \leq 9.8 \cdot 10^{-289}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ Vef KbT)) 1.0))))
   (if (<= Vef -3.55e+19)
     t_0
     (if (<= Vef -1.52e-226)
       (* 0.5 (+ NdChar NaChar))
       (if (<= Vef 9.8e-289)
         (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
         (if (<= Vef 8.5e+25) (/ 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 = NdChar / (exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -3.55e+19) {
		tmp = t_0;
	} else if (Vef <= -1.52e-226) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (Vef <= 9.8e-289) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else if (Vef <= 8.5e+25) {
		tmp = NdChar / (exp((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 = ndchar / (exp((vef / kbt)) + 1.0d0)
    if (vef <= (-3.55d+19)) then
        tmp = t_0
    else if (vef <= (-1.52d-226)) then
        tmp = 0.5d0 * (ndchar + nachar)
    else if (vef <= 9.8d-289) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    else if (vef <= 8.5d+25) then
        tmp = ndchar / (exp((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 = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -3.55e+19) {
		tmp = t_0;
	} else if (Vef <= -1.52e-226) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (Vef <= 9.8e-289) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else if (Vef <= 8.5e+25) {
		tmp = NdChar / (Math.exp((Ec / -KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((Vef / KbT)) + 1.0)
	tmp = 0
	if Vef <= -3.55e+19:
		tmp = t_0
	elif Vef <= -1.52e-226:
		tmp = 0.5 * (NdChar + NaChar)
	elif Vef <= 9.8e-289:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	elif Vef <= 8.5e+25:
		tmp = NdChar / (math.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(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0))
	tmp = 0.0
	if (Vef <= -3.55e+19)
		tmp = t_0;
	elseif (Vef <= -1.52e-226)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	elseif (Vef <= 9.8e-289)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	elseif (Vef <= 8.5e+25)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((Vef / KbT)) + 1.0);
	tmp = 0.0;
	if (Vef <= -3.55e+19)
		tmp = t_0;
	elseif (Vef <= -1.52e-226)
		tmp = 0.5 * (NdChar + NaChar);
	elseif (Vef <= 9.8e-289)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	elseif (Vef <= 8.5e+25)
		tmp = NdChar / (exp((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[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -3.55e+19], t$95$0, If[LessEqual[Vef, -1.52e-226], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 9.8e-289], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 8.5e+25], 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 Vef < -3.55e19 or 8.5000000000000007e25 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 56.5%

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

   if -3.55e19 < Vef < -1.52000000000000004e-226

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 35.7

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

   5. Applied rewrites 35.7%

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

   if -1.52000000000000004e-226 < Vef < 9.80000000000000016e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 57.9%

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

   if 9.80000000000000016e-289 < Vef < 8.5000000000000007e25

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in Ec around inf

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

   8. Applied rewrites 45.3%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -3.55 \cdot 10^{+19}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq -1.52 \cdot 10^{-226}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;Vef \leq 9.8 \cdot 10^{-289}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -2.55 \cdot 10^{-214}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 1.7 \cdot 10^{-252}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev + \left(Vef + EAccept\right)}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -2.55e-214)
   (/ NaChar (+ (exp (/ (- (+ Ev EAccept) mu) KbT)) 1.0))
   (if (<= NaChar 1.7e-252)
     (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
     (if (<= NaChar 2e-139)
       (/ NdChar (+ (exp (/ Ec (- KbT))) 1.0))
       (/ NaChar (+ (exp (/ (+ Ev (+ Vef EAccept)) KbT)) 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -2.55e-214) {
		tmp = NaChar / (exp((((Ev + EAccept) - mu) / KbT)) + 1.0);
	} else if (NaChar <= 1.7e-252) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else if (NaChar <= 2e-139) {
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp(((Ev + (Vef + EAccept)) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-2.55d-214)) then
        tmp = nachar / (exp((((ev + eaccept) - mu) / kbt)) + 1.0d0)
    else if (nachar <= 1.7d-252) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    else if (nachar <= 2d-139) then
        tmp = ndchar / (exp((ec / -kbt)) + 1.0d0)
    else
        tmp = nachar / (exp(((ev + (vef + eaccept)) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -2.55e-214) {
		tmp = NaChar / (Math.exp((((Ev + EAccept) - mu) / KbT)) + 1.0);
	} else if (NaChar <= 1.7e-252) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else if (NaChar <= 2e-139) {
		tmp = NdChar / (Math.exp((Ec / -KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp(((Ev + (Vef + EAccept)) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -2.55e-214:
		tmp = NaChar / (math.exp((((Ev + EAccept) - mu) / KbT)) + 1.0)
	elif NaChar <= 1.7e-252:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	elif NaChar <= 2e-139:
		tmp = NdChar / (math.exp((Ec / -KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp(((Ev + (Vef + EAccept)) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -2.55e-214)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) - mu) / KbT)) + 1.0));
	elseif (NaChar <= 1.7e-252)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	elseif (NaChar <= 2e-139)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Ev + Float64(Vef + EAccept)) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -2.55e-214)
		tmp = NaChar / (exp((((Ev + EAccept) - mu) / KbT)) + 1.0);
	elseif (NaChar <= 1.7e-252)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	elseif (NaChar <= 2e-139)
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	else
		tmp = NaChar / (exp(((Ev + (Vef + EAccept)) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -2.55e-214], N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.7e-252], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 2e-139], N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(Ev + N[(Vef + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -2.54999999999999993e-214

   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 mu around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate--l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in Vef around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 83.7

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

   8. Applied rewrites 83.7%

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

   9. Taylor expanded in NaChar around inf

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

   11. Applied rewrites 63.1%

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

   if -2.54999999999999993e-214 < NaChar < 1.7e-252

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 62.1%

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

   if 1.7e-252 < NaChar < 2.00000000000000006e-139

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in Ec around inf

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

   8. Applied rewrites 67.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}} \]

   if 2.00000000000000006e-139 < 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 mu around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate--l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in NaChar around inf

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

   8. Applied rewrites 62.4%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.55 \cdot 10^{-214}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 1.7 \cdot 10^{-252}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev + \left(Vef + EAccept\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{Ev + \left(Vef + EAccept\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -1 \cdot 10^{-213}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 1.7 \cdot 10^{-252}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ Ev (+ Vef EAccept)) KbT)) 1.0))))
   (if (<= NaChar -1e-213)
     t_0
     (if (<= NaChar 1.7e-252)
       (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
       (if (<= NaChar 2e-139) (/ 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 = NaChar / (exp(((Ev + (Vef + EAccept)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1e-213) {
		tmp = t_0;
	} else if (NaChar <= 1.7e-252) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else if (NaChar <= 2e-139) {
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp(((ev + (vef + eaccept)) / kbt)) + 1.0d0)
    if (nachar <= (-1d-213)) then
        tmp = t_0
    else if (nachar <= 1.7d-252) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    else if (nachar <= 2d-139) then
        tmp = ndchar / (exp((ec / -kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp(((Ev + (Vef + EAccept)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1e-213) {
		tmp = t_0;
	} else if (NaChar <= 1.7e-252) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else if (NaChar <= 2e-139) {
		tmp = NdChar / (Math.exp((Ec / -KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp(((Ev + (Vef + EAccept)) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -1e-213:
		tmp = t_0
	elif NaChar <= 1.7e-252:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	elif NaChar <= 2e-139:
		tmp = NdChar / (math.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(NaChar / Float64(exp(Float64(Float64(Ev + Float64(Vef + EAccept)) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -1e-213)
		tmp = t_0;
	elseif (NaChar <= 1.7e-252)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	elseif (NaChar <= 2e-139)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp(((Ev + (Vef + EAccept)) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -1e-213)
		tmp = t_0;
	elseif (NaChar <= 1.7e-252)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	elseif (NaChar <= 2e-139)
		tmp = NdChar / (exp((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[(Ev + N[(Vef + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1e-213], t$95$0, If[LessEqual[NaChar, 1.7e-252], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 2e-139], N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -9.9999999999999995e-214 or 2.00000000000000006e-139 < 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 mu around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate--l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in NaChar around inf

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

   8. Applied rewrites 62.0%

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

   if -9.9999999999999995e-214 < NaChar < 1.7e-252

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 62.1%

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

   if 1.7e-252 < NaChar < 2.00000000000000006e-139

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in Ec around inf

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

   8. Applied rewrites 67.9%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev + \left(Vef + EAccept\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 1.7 \cdot 10^{-252}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev + \left(Vef + EAccept\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -1.6 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 0.32:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu - Ec\right) + \left(Vef + EDonor\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)) 1.0))))
   (if (<= NaChar -1.6e-29)
     t_0
     (if (<= NaChar 0.32)
       (/ NdChar (+ (exp (/ (+ (- mu Ec) (+ Vef EDonor)) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.6e-29) {
		tmp = t_0;
	} else if (NaChar <= 0.32) {
		tmp = NdChar / (exp((((mu - Ec) + (Vef + EDonor)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp((((ev + eaccept) + (vef - mu)) / kbt)) + 1.0d0)
    if (nachar <= (-1.6d-29)) then
        tmp = t_0
    else if (nachar <= 0.32d0) then
        tmp = ndchar / (exp((((mu - ec) + (vef + edonor)) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.6e-29) {
		tmp = t_0;
	} else if (NaChar <= 0.32) {
		tmp = NdChar / (Math.exp((((mu - Ec) + (Vef + EDonor)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -1.6e-29:
		tmp = t_0
	elif NaChar <= 0.32:
		tmp = NdChar / (math.exp((((mu - Ec) + (Vef + 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(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -1.6e-29)
		tmp = t_0;
	elseif (NaChar <= 0.32)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu - Ec) + Float64(Vef + EDonor)) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -1.6e-29)
		tmp = t_0;
	elseif (NaChar <= 0.32)
		tmp = NdChar / (exp((((mu - Ec) + (Vef + EDonor)) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.6e-29], t$95$0, If[LessEqual[NaChar, 0.32], N[(NdChar / N[(N[Exp[N[(N[(N[(mu - Ec), $MachinePrecision] + N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.6e-29 or 0.320000000000000007 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -1.6e-29 < NaChar < 0.320000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 0.32:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu - Ec\right) + \left(Vef + EDonor\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -3.55 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq -1.52 \cdot 10^{-226}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;Vef \leq 3.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ Vef KbT)) 1.0))))
   (if (<= Vef -3.55e+19)
     t_0
     (if (<= Vef -1.52e-226)
       (* 0.5 (+ NdChar NaChar))
       (if (<= Vef 3.7e+26) (/ 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 = NdChar / (exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -3.55e+19) {
		tmp = t_0;
	} else if (Vef <= -1.52e-226) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (Vef <= 3.7e+26) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp((vef / kbt)) + 1.0d0)
    if (vef <= (-3.55d+19)) then
        tmp = t_0
    else if (vef <= (-1.52d-226)) then
        tmp = 0.5d0 * (ndchar + nachar)
    else if (vef <= 3.7d+26) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -3.55e+19) {
		tmp = t_0;
	} else if (Vef <= -1.52e-226) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (Vef <= 3.7e+26) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((Vef / KbT)) + 1.0)
	tmp = 0
	if Vef <= -3.55e+19:
		tmp = t_0
	elif Vef <= -1.52e-226:
		tmp = 0.5 * (NdChar + NaChar)
	elif Vef <= 3.7e+26:
		tmp = NdChar / (math.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(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0))
	tmp = 0.0
	if (Vef <= -3.55e+19)
		tmp = t_0;
	elseif (Vef <= -1.52e-226)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	elseif (Vef <= 3.7e+26)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((Vef / KbT)) + 1.0);
	tmp = 0.0;
	if (Vef <= -3.55e+19)
		tmp = t_0;
	elseif (Vef <= -1.52e-226)
		tmp = 0.5 * (NdChar + NaChar);
	elseif (Vef <= 3.7e+26)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -3.55e+19], t$95$0, If[LessEqual[Vef, -1.52e-226], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 3.7e+26], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -3.55e19 or 3.69999999999999988e26 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 56.5%

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

   if -3.55e19 < Vef < -1.52000000000000004e-226

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 35.7

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

   5. Applied rewrites 35.7%

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

   if -1.52000000000000004e-226 < Vef < 3.69999999999999988e26

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 41.3%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -3.55 \cdot 10^{+19}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq -1.52 \cdot 10^{-226}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;Vef \leq 3.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 22.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -8.5 \cdot 10^{-128}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -8.5e-128)
   (* NaChar 0.5)
   (if (<= NaChar 2.6e-139) (* 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 <= -8.5e-128) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 2.6e-139) {
		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 <= (-8.5d-128)) then
        tmp = nachar * 0.5d0
    else if (nachar <= 2.6d-139) 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 <= -8.5e-128) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 2.6e-139) {
		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 <= -8.5e-128:
		tmp = NaChar * 0.5
	elif NaChar <= 2.6e-139:
		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 <= -8.5e-128)
		tmp = Float64(NaChar * 0.5);
	elseif (NaChar <= 2.6e-139)
		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 <= -8.5e-128)
		tmp = NaChar * 0.5;
	elseif (NaChar <= 2.6e-139)
		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, -8.5e-128], N[(NaChar * 0.5), $MachinePrecision], If[LessEqual[NaChar, 2.6e-139], N[(NdChar * 0.5), $MachinePrecision], N[(NaChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -8.4999999999999996e-128 or 2.5999999999999998e-139 < 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. lower-*.f64 N/A

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

      3. lower-+.f64 25.4

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

   5. Applied rewrites 25.4%

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

   6. Taylor expanded in NaChar around inf

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

   8. Applied rewrites 21.7%

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

   if -8.4999999999999996e-128 < NaChar < 2.5999999999999998e-139

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. 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. lower-+.f64 N/A

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

      11. lower-+.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. lower--.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 26.3%

      \[\leadsto NdChar \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 27.5% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(NdChar + NaChar\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 25.9

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

5. Applied rewrites 25.9%

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

6. Final simplification 25.9%

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

Alternative 16: 18.7% accurate, 46.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 25.9

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

5. Applied rewrites 25.9%

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

6. Taylor expanded in NaChar around inf

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

8. Applied rewrites 17.0%

   \[\leadsto NaChar \cdot \color{blue}{0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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))))))