Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 17.5s

Alternatives: 25

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = -2.496776807412835e-195
  2. Ec = 1.8127470901785636e-47
  3. Vef = -6.699845053240453e-247
  4. EDonor = 1.2576467949731456e+236
  5. mu = 6.152310546792556e-98
  6. KbT = -2.482449763318707e+34
  7. NaChar = -3.933991773974218e-195
  8. Ev = 2.3095320526510408e+288
  9. EAccept = -1.3212621467523892e+197

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

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_1 := e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}\\ t_2 := \frac{NaChar}{e^{\frac{Vef}{KbT}} - -1} - \frac{NdChar}{-1 - t\_1}\\ t_3 := \frac{NdChar}{t\_1 - -1} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-185}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{NaChar}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_1 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)))
        (t_2 (- (/ NaChar (- (exp (/ Vef KbT)) -1.0)) (/ NdChar (- -1.0 t_1))))
        (t_3 (- (/ NdChar (- t_1 -1.0)) (/ NaChar (- -1.0 t_0)))))
   (if (<= t_3 -2e-185) t_2 (if (<= t_3 0.0) (/ NaChar (- t_0 -1.0)) 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 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	double t_2 = (NaChar / (exp((Vef / KbT)) - -1.0)) - (NdChar / (-1.0 - t_1));
	double t_3 = (NdChar / (t_1 - -1.0)) - (NaChar / (-1.0 - t_0));
	double tmp;
	if (t_3 <= -2e-185) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = NaChar / (t_0 - -1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_1 = exp(((mu - ((ec - vef) - edonor)) / kbt))
    t_2 = (nachar / (exp((vef / kbt)) - (-1.0d0))) - (ndchar / ((-1.0d0) - t_1))
    t_3 = (ndchar / (t_1 - (-1.0d0))) - (nachar / ((-1.0d0) - t_0))
    if (t_3 <= (-2d-185)) then
        tmp = t_2
    else if (t_3 <= 0.0d0) then
        tmp = nachar / (t_0 - (-1.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	double t_2 = (NaChar / (Math.exp((Vef / KbT)) - -1.0)) - (NdChar / (-1.0 - t_1));
	double t_3 = (NdChar / (t_1 - -1.0)) - (NaChar / (-1.0 - t_0));
	double tmp;
	if (t_3 <= -2e-185) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = NaChar / (t_0 - -1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))
	t_2 = (NaChar / (math.exp((Vef / KbT)) - -1.0)) - (NdChar / (-1.0 - t_1))
	t_3 = (NdChar / (t_1 - -1.0)) - (NaChar / (-1.0 - t_0))
	tmp = 0
	if t_3 <= -2e-185:
		tmp = t_2
	elif t_3 <= 0.0:
		tmp = NaChar / (t_0 - -1.0)
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT))
	t_2 = Float64(Float64(NaChar / Float64(exp(Float64(Vef / KbT)) - -1.0)) - Float64(NdChar / Float64(-1.0 - t_1)))
	t_3 = Float64(Float64(NdChar / Float64(t_1 - -1.0)) - Float64(NaChar / Float64(-1.0 - t_0)))
	tmp = 0.0
	if (t_3 <= -2e-185)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(NaChar / Float64(t_0 - -1.0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_1 = exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	t_2 = (NaChar / (exp((Vef / KbT)) - -1.0)) - (NdChar / (-1.0 - t_1));
	t_3 = (NdChar / (t_1 - -1.0)) - (NaChar / (-1.0 - t_0));
	tmp = 0.0;
	if (t_3 <= -2e-185)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = NaChar / (t_0 - -1.0);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-185], t$95$2, If[LessEqual[t$95$3, 0.0], N[(NaChar / N[(t$95$0 - -1.0), $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))))) < -2e-185 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 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 Vef around inf

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

      1. lower-/.f64 77.9

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

   5. Applied rewrites 77.9%

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

   if -2e-185 < (+.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 NaChar around inf

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 96.5

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

   5. Applied rewrites 96.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-185}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} - -1} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} - -1} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-127}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{Ec - \left(\left(mu + Vef\right) + EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{NdChar - NaChar}{NaChar + NdChar}} \cdot \left(NdChar - NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NdChar (- (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) -1.0))
          (/ NaChar (- -1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (<= t_0 -1e-127)
     (* (+ NaChar NdChar) 0.5)
     (if (<= t_0 5e-240)
       (/ NdChar (- 2.0 (/ (- Ec (+ (+ mu Vef) EDonor)) KbT)))
       (*
        (/ 0.5 (/ (- NdChar NaChar) (+ NaChar NdChar)))
        (- NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_0 <= -1e-127) {
		tmp = (NaChar + NdChar) * 0.5;
	} else if (t_0 <= 5e-240) {
		tmp = NdChar / (2.0 - ((Ec - ((mu + Vef) + EDonor)) / KbT));
	} else {
		tmp = (0.5 / ((NdChar - NaChar) / (NaChar + NdChar))) * (NdChar - NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp(((mu - ((ec - vef) - edonor)) / kbt)) - (-1.0d0))) - (nachar / ((-1.0d0) - exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if (t_0 <= (-1d-127)) then
        tmp = (nachar + ndchar) * 0.5d0
    else if (t_0 <= 5d-240) then
        tmp = ndchar / (2.0d0 - ((ec - ((mu + vef) + edonor)) / kbt))
    else
        tmp = (0.5d0 / ((ndchar - nachar) / (nachar + ndchar))) * (ndchar - nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_0 <= -1e-127) {
		tmp = (NaChar + NdChar) * 0.5;
	} else if (t_0 <= 5e-240) {
		tmp = NdChar / (2.0 - ((Ec - ((mu + Vef) + EDonor)) / KbT));
	} else {
		tmp = (0.5 / ((NdChar - NaChar) / (NaChar + NdChar))) * (NdChar - NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_0 <= -1e-127:
		tmp = (NaChar + NdChar) * 0.5
	elif t_0 <= 5e-240:
		tmp = NdChar / (2.0 - ((Ec - ((mu + Vef) + EDonor)) / KbT))
	else:
		tmp = (0.5 / ((NdChar - NaChar) / (NaChar + NdChar))) * (NdChar - NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)) - -1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_0 <= -1e-127)
		tmp = Float64(Float64(NaChar + NdChar) * 0.5);
	elseif (t_0 <= 5e-240)
		tmp = Float64(NdChar / Float64(2.0 - Float64(Float64(Ec - Float64(Float64(mu + Vef) + EDonor)) / KbT)));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(NdChar - NaChar) / Float64(NaChar + NdChar))) * Float64(NdChar - NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_0 <= -1e-127)
		tmp = (NaChar + NdChar) * 0.5;
	elseif (t_0 <= 5e-240)
		tmp = NdChar / (2.0 - ((Ec - ((mu + Vef) + EDonor)) / KbT));
	else
		tmp = (0.5 / ((NdChar - NaChar) / (NaChar + NdChar))) * (NdChar - NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $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]}, If[LessEqual[t$95$0, -1e-127], N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 5e-240], N[(NdChar / N[(2.0 - N[(N[(Ec - N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(NdChar - NaChar), $MachinePrecision] / N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(NdChar - NaChar), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1e-127

   1. Initial program 99.7%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 36.6

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

   5. Applied rewrites 36.6%

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

   if -1e-127 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5.0000000000000004e-240

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in KbT around -inf

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

   11. Applied rewrites 46.4%

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

   if 5.0000000000000004e-240 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.1

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

   5. Applied rewrites 37.1%

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

   7. Applied rewrites 17.8%

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

   9. Applied rewrites 37.1%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-127}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-240}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{Ec - \left(\left(mu + Vef\right) + EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{NdChar - NaChar}{NaChar + NdChar}} \cdot \left(NdChar - NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-200}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-249}:\\ \;\;\;\;\frac{NdChar}{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{NdChar - NaChar}{NaChar + NdChar}} \cdot \left(NdChar - NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NdChar (- (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) -1.0))
          (/ NaChar (- -1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (<= t_0 -5e-200)
     (* (+ NaChar NdChar) 0.5)
     (if (<= t_0 5e-249)
       (/ NdChar (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))
       (*
        (/ 0.5 (/ (- NdChar NaChar) (+ NaChar NdChar)))
        (- NdChar NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_0 <= -5e-200) {
		tmp = (NaChar + NdChar) * 0.5;
	} else if (t_0 <= 5e-249) {
		tmp = NdChar / ((((mu + Vef) + EDonor) - Ec) / KbT);
	} else {
		tmp = (0.5 / ((NdChar - NaChar) / (NaChar + NdChar))) * (NdChar - NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (exp(((mu - ((ec - vef) - edonor)) / kbt)) - (-1.0d0))) - (nachar / ((-1.0d0) - exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if (t_0 <= (-5d-200)) then
        tmp = (nachar + ndchar) * 0.5d0
    else if (t_0 <= 5d-249) then
        tmp = ndchar / ((((mu + vef) + edonor) - ec) / kbt)
    else
        tmp = (0.5d0 / ((ndchar - nachar) / (nachar + ndchar))) * (ndchar - nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_0 <= -5e-200) {
		tmp = (NaChar + NdChar) * 0.5;
	} else if (t_0 <= 5e-249) {
		tmp = NdChar / ((((mu + Vef) + EDonor) - Ec) / KbT);
	} else {
		tmp = (0.5 / ((NdChar - NaChar) / (NaChar + NdChar))) * (NdChar - NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_0 <= -5e-200:
		tmp = (NaChar + NdChar) * 0.5
	elif t_0 <= 5e-249:
		tmp = NdChar / ((((mu + Vef) + EDonor) - Ec) / KbT)
	else:
		tmp = (0.5 / ((NdChar - NaChar) / (NaChar + NdChar))) * (NdChar - NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)) - -1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_0 <= -5e-200)
		tmp = Float64(Float64(NaChar + NdChar) * 0.5);
	elseif (t_0 <= 5e-249)
		tmp = Float64(NdChar / Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(NdChar - NaChar) / Float64(NaChar + NdChar))) * Float64(NdChar - NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_0 <= -5e-200)
		tmp = (NaChar + NdChar) * 0.5;
	elseif (t_0 <= 5e-249)
		tmp = NdChar / ((((mu + Vef) + EDonor) - Ec) / KbT);
	else
		tmp = (0.5 / ((NdChar - NaChar) / (NaChar + NdChar))) * (NdChar - NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $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]}, If[LessEqual[t$95$0, -5e-200], N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 5e-249], N[(NdChar / N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(NdChar - NaChar), $MachinePrecision] / N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(NdChar - NaChar), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.99999999999999991e-200

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 35.5

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

   5. Applied rewrites 35.5%

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

   if -4.99999999999999991e-200 < (+.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-249

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 41.6%

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

   9. Taylor expanded in KbT around 0

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

   11. Applied rewrites 48.5%

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

   if 4.9999999999999999e-249 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.3

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

   5. Applied rewrites 37.3%

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

   7. Applied rewrites 17.5%

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

   9. Applied rewrites 37.3%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -5 \cdot 10^{-200}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-249}:\\ \;\;\;\;\frac{NdChar}{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{NdChar - NaChar}{NaChar + NdChar}} \cdot \left(NdChar - NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-200}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-249}:\\ \;\;\;\;\frac{NdChar}{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (- (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) -1.0))
          (/ NaChar (- -1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (<= t_1 -5e-200)
     t_0
     (if (<= t_1 5e-249)
       (/ NdChar (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -5e-200) {
		tmp = t_0;
	} else if (t_1 <= 5e-249) {
		tmp = NdChar / ((((mu + Vef) + EDonor) - Ec) / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (exp(((mu - ((ec - vef) - edonor)) / kbt)) - (-1.0d0))) - (nachar / ((-1.0d0) - exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if (t_1 <= (-5d-200)) then
        tmp = t_0
    else if (t_1 <= 5d-249) then
        tmp = ndchar / ((((mu + vef) + edonor) - ec) / kbt)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -5e-200) {
		tmp = t_0;
	} else if (t_1 <= 5e-249) {
		tmp = NdChar / ((((mu + Vef) + EDonor) - Ec) / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -5e-200:
		tmp = t_0
	elif t_1 <= 5e-249:
		tmp = NdChar / ((((mu + Vef) + EDonor) - Ec) / KbT)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)) - -1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -5e-200)
		tmp = t_0;
	elseif (t_1 <= 5e-249)
		tmp = Float64(NdChar / Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -5e-200)
		tmp = t_0;
	elseif (t_1 <= 5e-249)
		tmp = NdChar / ((((mu + Vef) + EDonor) - Ec) / KbT);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $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[(-1.0 - N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-200], t$95$0, If[LessEqual[t$95$1, 5e-249], N[(NdChar / N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.99999999999999991e-200 or 4.9999999999999999e-249 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 36.4

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

   5. Applied rewrites 36.4%

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

   if -4.99999999999999991e-200 < (+.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-249

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 41.6%

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

   9. Taylor expanded in KbT around 0

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

   11. Applied rewrites 48.5%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -5 \cdot 10^{-200}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-249}:\\ \;\;\;\;\frac{NdChar}{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-260}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-249}:\\ \;\;\;\;\frac{-0.5 \cdot \left(NdChar \cdot NdChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (- (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) -1.0))
          (/ NaChar (- -1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (<= t_1 -2e-260)
     t_0
     (if (<= t_1 5e-249)
       (/ (* -0.5 (* NdChar NdChar)) (- NaChar NdChar))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-260) {
		tmp = t_0;
	} else if (t_1 <= 5e-249) {
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (exp(((mu - ((ec - vef) - edonor)) / kbt)) - (-1.0d0))) - (nachar / ((-1.0d0) - exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if (t_1 <= (-2d-260)) then
        tmp = t_0
    else if (t_1 <= 5d-249) then
        tmp = ((-0.5d0) * (ndchar * ndchar)) / (nachar - ndchar)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-260) {
		tmp = t_0;
	} else if (t_1 <= 5e-249) {
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -2e-260:
		tmp = t_0
	elif t_1 <= 5e-249:
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)) - -1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-260)
		tmp = t_0;
	elseif (t_1 <= 5e-249)
		tmp = Float64(Float64(-0.5 * Float64(NdChar * NdChar)) / Float64(NaChar - NdChar));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -2e-260)
		tmp = t_0;
	elseif (t_1 <= 5e-249)
		tmp = (-0.5 * (NdChar * NdChar)) / (NaChar - NdChar);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $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[(-1.0 - N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-260], t$95$0, If[LessEqual[t$95$1, 5e-249], N[(N[(-0.5 * N[(NdChar * NdChar), $MachinePrecision]), $MachinePrecision] / N[(NaChar - NdChar), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.99999999999999992e-260 or 4.9999999999999999e-249 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 35.4

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

   5. Applied rewrites 35.4%

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

   if -1.99999999999999992e-260 < (+.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-249

   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 2.8

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

   5. Applied rewrites 2.8%

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

   7. Applied rewrites 4.5%

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

   8. Taylor expanded in NaChar around 0

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

   10. Applied rewrites 31.6%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-260}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-249}:\\ \;\;\;\;\frac{-0.5 \cdot \left(NdChar \cdot NdChar\right)}{NaChar - NdChar}\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-265}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-269}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (- (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) -1.0))
          (/ NaChar (- -1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (<= t_1 -1e-265) t_0 (if (<= t_1 5e-269) (/ NdChar (/ Vef KbT)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-265) {
		tmp = t_0;
	} else if (t_1 <= 5e-269) {
		tmp = NdChar / (Vef / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (exp(((mu - ((ec - vef) - edonor)) / kbt)) - (-1.0d0))) - (nachar / ((-1.0d0) - exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if (t_1 <= (-1d-265)) then
        tmp = t_0
    else if (t_1 <= 5d-269) then
        tmp = ndchar / (vef / kbt)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-265) {
		tmp = t_0;
	} else if (t_1 <= 5e-269) {
		tmp = NdChar / (Vef / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -1e-265:
		tmp = t_0
	elif t_1 <= 5e-269:
		tmp = NdChar / (Vef / KbT)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)) - -1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -1e-265)
		tmp = t_0;
	elseif (t_1 <= 5e-269)
		tmp = Float64(NdChar / Float64(Vef / KbT));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -1e-265)
		tmp = t_0;
	elseif (t_1 <= 5e-269)
		tmp = NdChar / (Vef / KbT);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $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[(-1.0 - N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-265], t$95$0, If[LessEqual[t$95$1, 5e-269], N[(NdChar / N[(Vef / KbT), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999985e-266 or 4.99999999999999979e-269 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 34.8

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

   5. Applied rewrites 34.8%

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

   if -9.99999999999999985e-266 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999979e-269

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 49.9%

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

   9. Taylor expanded in Vef around inf

      \[\leadsto \frac{NdChar}{\frac{Vef}{KbT}} \]
   10. Step-by-step derivation

   11. Applied rewrites 28.5%

       \[\leadsto \frac{NdChar}{\frac{Vef}{KbT}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-265}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-269}:\\ \;\;\;\;\frac{NdChar}{\frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-217}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-262}:\\ \;\;\;\;\frac{NdChar}{\frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (- (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)) -1.0))
          (/ NaChar (- -1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (<= t_1 -1e-217)
     t_0
     (if (<= t_1 4e-262) (/ NdChar (/ EDonor KbT)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-217) {
		tmp = t_0;
	} else if (t_1 <= 4e-262) {
		tmp = NdChar / (EDonor / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (exp(((mu - ((ec - vef) - edonor)) / kbt)) - (-1.0d0))) - (nachar / ((-1.0d0) - exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if (t_1 <= (-1d-217)) then
        tmp = t_0
    else if (t_1 <= 4d-262) then
        tmp = ndchar / (edonor / kbt)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-217) {
		tmp = t_0;
	} else if (t_1 <= 4e-262) {
		tmp = NdChar / (EDonor / KbT);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -1e-217:
		tmp = t_0
	elif t_1 <= 4e-262:
		tmp = NdChar / (EDonor / KbT)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)) - -1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -1e-217)
		tmp = t_0;
	elseif (t_1 <= 4e-262)
		tmp = Float64(NdChar / 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 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (NaChar / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -1e-217)
		tmp = t_0;
	elseif (t_1 <= 4e-262)
		tmp = NdChar / (EDonor / KbT);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $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[(-1.0 - N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-217], t$95$0, If[LessEqual[t$95$1, 4e-262], N[(NdChar / N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.00000000000000008e-217 or 4.00000000000000005e-262 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 35.8

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

   5. Applied rewrites 35.8%

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

   if -1.00000000000000008e-217 < (+.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))))) < 4.00000000000000005e-262

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 44.2%

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

   9. Taylor expanded in EDonor around inf

      \[\leadsto \frac{NdChar}{\frac{EDonor}{KbT}} \]
   10. Step-by-step derivation

   11. Applied rewrites 22.0%

       \[\leadsto \frac{NdChar}{\frac{EDonor}{KbT}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}} - -1} - \frac{NaChar}{-1 - e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 4 \cdot 10^{-262}:\\ \;\;\;\;\frac{NdChar}{\frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

Python

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. distribute-frac-neg N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 10: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / (math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)) - -1.0)) - (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(exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)) - -1.0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
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 / (-1.0 - exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
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[(-1.0 - N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Final simplification 99.9%

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

Alternative 11: 69.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(mu + Vef\right) + EDonor\right) - Ec\\ \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{1}{\frac{KbT}{t\_0}}} - -1}\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{t\_0}{KbT}} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (+ (+ mu Vef) EDonor) Ec)))
   (if (<= NdChar -1.6e+63)
     (/ NdChar (- (exp (/ 1.0 (/ KbT t_0))) -1.0))
     (if (<= NdChar 1.75e+101)
       (/ NaChar (- (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) -1.0))
       (/ NdChar (- (exp (/ t_0 KbT)) -1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = ((mu + Vef) + EDonor) - Ec;
	double tmp;
	if (NdChar <= -1.6e+63) {
		tmp = NdChar / (exp((1.0 / (KbT / t_0))) - -1.0);
	} else if (NdChar <= 1.75e+101) {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0);
	} else {
		tmp = NdChar / (exp((t_0 / KbT)) - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((mu + vef) + edonor) - ec
    if (ndchar <= (-1.6d+63)) then
        tmp = ndchar / (exp((1.0d0 / (kbt / t_0))) - (-1.0d0))
    else if (ndchar <= 1.75d+101) then
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) - (-1.0d0))
    else
        tmp = ndchar / (exp((t_0 / kbt)) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = ((mu + Vef) + EDonor) - Ec;
	double tmp;
	if (NdChar <= -1.6e+63) {
		tmp = NdChar / (Math.exp((1.0 / (KbT / t_0))) - -1.0);
	} else if (NdChar <= 1.75e+101) {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0);
	} else {
		tmp = NdChar / (Math.exp((t_0 / KbT)) - -1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = ((mu + Vef) + EDonor) - Ec
	tmp = 0
	if NdChar <= -1.6e+63:
		tmp = NdChar / (math.exp((1.0 / (KbT / t_0))) - -1.0)
	elif NdChar <= 1.75e+101:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0)
	else:
		tmp = NdChar / (math.exp((t_0 / KbT)) - -1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Float64(mu + Vef) + EDonor) - Ec)
	tmp = 0.0
	if (NdChar <= -1.6e+63)
		tmp = Float64(NdChar / Float64(exp(Float64(1.0 / Float64(KbT / t_0))) - -1.0));
	elseif (NdChar <= 1.75e+101)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(t_0 / KbT)) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = ((mu + Vef) + EDonor) - Ec;
	tmp = 0.0;
	if (NdChar <= -1.6e+63)
		tmp = NdChar / (exp((1.0 / (KbT / t_0))) - -1.0);
	elseif (NdChar <= 1.75e+101)
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0);
	else
		tmp = NdChar / (exp((t_0 / KbT)) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision]}, If[LessEqual[NdChar, -1.6e+63], N[(NdChar / N[(N[Exp[N[(1.0 / N[(KbT / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.75e+101], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(t$95$0 / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.60000000000000006e63

   1. Initial program 99.6%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 78.4%

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

   if -1.60000000000000006e63 < NdChar < 1.75000000000000012e101

   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 NaChar around inf

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 1.75000000000000012e101 < NdChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 69.6

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

   5. Applied rewrites 69.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{1}{\frac{KbT}{\left(\left(mu + Vef\right) + EDonor\right) - Ec}}} - -1}\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\\ \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{NdChar}{{\mathsf{E}\left(\right)}^{t\_0} - -1}\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{t\_0} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)))
   (if (<= NdChar -1.6e+63)
     (/ NdChar (- (pow (E) t_0) -1.0))
     (if (<= NdChar 1.75e+101)
       (/ NaChar (- (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) -1.0))
       (/ NdChar (- (exp t_0) -1.0))))))

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.60000000000000006e63

   1. Initial program 99.6%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 78.3%

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

   if -1.60000000000000006e63 < NdChar < 1.75000000000000012e101

   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 NaChar around inf

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 1.75000000000000012e101 < NdChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 69.6

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

   5. Applied rewrites 69.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{NdChar}{{\mathsf{E}\left(\right)}^{\left(\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right)} - -1}\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} - -1}\\ \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) -1.0))))
   (if (<= NdChar -1.6e+63)
     t_0
     (if (<= NdChar 1.75e+101)
       (/ NaChar (- (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) -1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -1.6e+63) {
		tmp = t_0;
	} else if (NdChar <= 1.75e+101) {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) - (-1.0d0))
    if (ndchar <= (-1.6d+63)) then
        tmp = t_0
    else if (ndchar <= 1.75d+101) then
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -1.6e+63) {
		tmp = t_0;
	} else if (NdChar <= 1.75e+101) {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) - -1.0)
	tmp = 0
	if NdChar <= -1.6e+63:
		tmp = t_0
	elif NdChar <= 1.75e+101:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) - -1.0))
	tmp = 0.0
	if (NdChar <= -1.6e+63)
		tmp = t_0;
	elseif (NdChar <= 1.75e+101)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) - -1.0);
	tmp = 0.0;
	if (NdChar <= -1.6e+63)
		tmp = t_0;
	elseif (NdChar <= 1.75e+101)
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.6e+63], t$95$0, If[LessEqual[NdChar, 1.75e+101], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.60000000000000006e63 or 1.75000000000000012e101 < NdChar

   1. Initial program 99.8%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 74.6

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

   5. Applied rewrites 74.6%

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

   if -1.60000000000000006e63 < NdChar < 1.75000000000000012e101

   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 NaChar around inf

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 72.0

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

   5. Applied rewrites 72.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} - -1}\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} - -1}\\ \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 1.82 \cdot 10^{+140}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- (exp (/ (- (+ EDonor Vef) Ec) KbT)) -1.0))))
   (if (<= NdChar -1.6e+63)
     t_0
     (if (<= NdChar 1.82e+140)
       (/ NaChar (- (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) -1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -1.6e+63) {
		tmp = t_0;
	} else if (NdChar <= 1.82e+140) {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp((((edonor + vef) - ec) / kbt)) - (-1.0d0))
    if (ndchar <= (-1.6d+63)) then
        tmp = t_0
    else if (ndchar <= 1.82d+140) then
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -1.6e+63) {
		tmp = t_0;
	} else if (NdChar <= 1.82e+140) {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) - -1.0)
	tmp = 0
	if NdChar <= -1.6e+63:
		tmp = t_0
	elif NdChar <= 1.82e+140:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) - -1.0))
	tmp = 0.0
	if (NdChar <= -1.6e+63)
		tmp = t_0;
	elseif (NdChar <= 1.82e+140)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) - -1.0);
	tmp = 0.0;
	if (NdChar <= -1.6e+63)
		tmp = t_0;
	elseif (NdChar <= 1.82e+140)
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.6e+63], t$95$0, If[LessEqual[NdChar, 1.82e+140], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.60000000000000006e63 or 1.82e140 < NdChar

   1. Initial program 99.8%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in mu around 0

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

   8. Applied rewrites 72.6%

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

   if -1.60000000000000006e63 < NdChar < 1.82e140

   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 NaChar around inf

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 71.5

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

   5. Applied rewrites 71.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} - -1}\\ \mathbf{elif}\;NdChar \leq 1.82 \cdot 10^{+140}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot NaChar - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{if}\;KbT \leq -3.8 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.7 \cdot 10^{+179}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (* 0.5 NaChar) (/ NdChar (- -1.0 (exp (/ EDonor KbT)))))))
   (if (<= KbT -3.8e+121)
     t_0
     (if (<= KbT 2.7e+179)
       (/ 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 = (0.5 * NaChar) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	double tmp;
	if (KbT <= -3.8e+121) {
		tmp = t_0;
	} else if (KbT <= 2.7e+179) {
		tmp = 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 = (0.5d0 * nachar) - (ndchar / ((-1.0d0) - exp((edonor / kbt))))
    if (kbt <= (-3.8d+121)) then
        tmp = t_0
    else if (kbt <= 2.7d+179) then
        tmp = 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 = (0.5 * NaChar) - (NdChar / (-1.0 - Math.exp((EDonor / KbT))));
	double tmp;
	if (KbT <= -3.8e+121) {
		tmp = t_0;
	} else if (KbT <= 2.7e+179) {
		tmp = 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 = (0.5 * NaChar) - (NdChar / (-1.0 - math.exp((EDonor / KbT))))
	tmp = 0
	if KbT <= -3.8e+121:
		tmp = t_0
	elif KbT <= 2.7e+179:
		tmp = 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(0.5 * NaChar) - Float64(NdChar / Float64(-1.0 - exp(Float64(EDonor / KbT)))))
	tmp = 0.0
	if (KbT <= -3.8e+121)
		tmp = t_0;
	elseif (KbT <= 2.7e+179)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + 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 = (0.5 * NaChar) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	tmp = 0.0;
	if (KbT <= -3.8e+121)
		tmp = t_0;
	elseif (KbT <= 2.7e+179)
		tmp = 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[(0.5 * NaChar), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -3.8e+121], t$95$0, If[LessEqual[KbT, 2.7e+179], N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -3.8e121 or 2.69999999999999982e179 < KbT

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 70.6

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

   8. Applied rewrites 70.6%

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

   if -3.8e121 < KbT < 2.69999999999999982e179

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in mu around 0

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

   8. Applied rewrites 54.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.8 \cdot 10^{+121}:\\ \;\;\;\;0.5 \cdot NaChar - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;KbT \leq 2.7 \cdot 10^{+179}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NaChar - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{if}\;KbT \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{NaChar}{KbT} \cdot Ev, t\_0\right)\\ \mathbf{elif}\;KbT \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} - -1}\\ \mathbf{elif}\;KbT \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{-NdChar}{KbT} \cdot Ec, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5)))
   (if (<= KbT -5.2e+131)
     (fma -0.25 (* (/ NaChar KbT) Ev) t_0)
     (if (<= KbT -2.2e+29)
       (/ NdChar (- (exp (/ mu KbT)) -1.0))
       (if (<= KbT 3.5e+69)
         (/ NdChar (- (exp (/ EDonor KbT)) -1.0))
         (fma -0.25 (* (/ (- NdChar) KbT) Ec) t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double tmp;
	if (KbT <= -5.2e+131) {
		tmp = fma(-0.25, ((NaChar / KbT) * Ev), t_0);
	} else if (KbT <= -2.2e+29) {
		tmp = NdChar / (exp((mu / KbT)) - -1.0);
	} else if (KbT <= 3.5e+69) {
		tmp = NdChar / (exp((EDonor / KbT)) - -1.0);
	} else {
		tmp = fma(-0.25, ((-NdChar / KbT) * Ec), t_0);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	tmp = 0.0
	if (KbT <= -5.2e+131)
		tmp = fma(-0.25, Float64(Float64(NaChar / KbT) * Ev), t_0);
	elseif (KbT <= -2.2e+29)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) - -1.0));
	elseif (KbT <= 3.5e+69)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) - -1.0));
	else
		tmp = fma(-0.25, Float64(Float64(Float64(-NdChar) / KbT) * Ec), t_0);
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[KbT, -5.2e+131], N[(-0.25 * N[(N[(NaChar / KbT), $MachinePrecision] * Ev), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[KbT, -2.2e+29], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 3.5e+69], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[((-NdChar) / KbT), $MachinePrecision] * Ec), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -5.2e131

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-neg-in N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in Ev around inf

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

   8. Applied rewrites 61.5%

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

   if -5.2e131 < KbT < -2.2000000000000001e29

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in mu around inf

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

   8. Applied rewrites 49.9%

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

   if -2.2000000000000001e29 < KbT < 3.49999999999999987e69

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 38.2%

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

   if 3.49999999999999987e69 < KbT

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-neg-in N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in Ec around inf

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

   8. Applied rewrites 52.1%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{NaChar}{KbT} \cdot Ev, \left(NaChar + NdChar\right) \cdot 0.5\right)\\ \mathbf{elif}\;KbT \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} - -1}\\ \mathbf{elif}\;KbT \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{-NdChar}{KbT} \cdot Ec, \left(NaChar + NdChar\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 40.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{if}\;KbT \leq -6 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{NaChar}{KbT} \cdot Ev, t\_0\right)\\ \mathbf{elif}\;KbT \leq -1.35 \cdot 10^{+44}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} - -1}\\ \mathbf{elif}\;KbT \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{-NdChar}{KbT} \cdot Ec, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5)))
   (if (<= KbT -6e+131)
     (fma -0.25 (* (/ NaChar KbT) Ev) t_0)
     (if (<= KbT -1.35e+44)
       (/ NdChar (- (exp (/ Vef KbT)) -1.0))
       (if (<= KbT 3.5e+69)
         (/ NdChar (- (exp (/ EDonor KbT)) -1.0))
         (fma -0.25 (* (/ (- NdChar) KbT) Ec) t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double tmp;
	if (KbT <= -6e+131) {
		tmp = fma(-0.25, ((NaChar / KbT) * Ev), t_0);
	} else if (KbT <= -1.35e+44) {
		tmp = NdChar / (exp((Vef / KbT)) - -1.0);
	} else if (KbT <= 3.5e+69) {
		tmp = NdChar / (exp((EDonor / KbT)) - -1.0);
	} else {
		tmp = fma(-0.25, ((-NdChar / KbT) * Ec), t_0);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	tmp = 0.0
	if (KbT <= -6e+131)
		tmp = fma(-0.25, Float64(Float64(NaChar / KbT) * Ev), t_0);
	elseif (KbT <= -1.35e+44)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) - -1.0));
	elseif (KbT <= 3.5e+69)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) - -1.0));
	else
		tmp = fma(-0.25, Float64(Float64(Float64(-NdChar) / KbT) * Ec), t_0);
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[KbT, -6e+131], N[(-0.25 * N[(N[(NaChar / KbT), $MachinePrecision] * Ev), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[KbT, -1.35e+44], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 3.5e+69], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[((-NdChar) / KbT), $MachinePrecision] * Ec), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -6.0000000000000003e131

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-neg-in N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in Ev around inf

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

   8. Applied rewrites 61.5%

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

   if -6.0000000000000003e131 < KbT < -1.35e44

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 46.7%

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

   if -1.35e44 < KbT < 3.49999999999999987e69

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 38.8%

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

   if 3.49999999999999987e69 < KbT

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-neg-in N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in Ec around inf

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

   8. Applied rewrites 52.1%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{NaChar}{KbT} \cdot Ev, \left(NaChar + NdChar\right) \cdot 0.5\right)\\ \mathbf{elif}\;KbT \leq -1.35 \cdot 10^{+44}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} - -1}\\ \mathbf{elif}\;KbT \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{-NdChar}{KbT} \cdot Ec, \left(NaChar + NdChar\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} - -1}\\ \mathbf{if}\;NdChar \leq -9.8 \cdot 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 1.65 \cdot 10^{+101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- (exp (/ (- (+ EDonor Vef) Ec) KbT)) -1.0))))
   (if (<= NdChar -9.8e-58)
     t_0
     (if (<= NdChar 1.65e+101) (/ NaChar (- (exp (/ Vef KbT)) -1.0)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -9.8e-58) {
		tmp = t_0;
	} else if (NdChar <= 1.65e+101) {
		tmp = NaChar / (exp((Vef / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp((((edonor + vef) - ec) / kbt)) - (-1.0d0))
    if (ndchar <= (-9.8d-58)) then
        tmp = t_0
    else if (ndchar <= 1.65d+101) then
        tmp = nachar / (exp((vef / kbt)) - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -9.8e-58) {
		tmp = t_0;
	} else if (NdChar <= 1.65e+101) {
		tmp = NaChar / (Math.exp((Vef / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) - -1.0)
	tmp = 0
	if NdChar <= -9.8e-58:
		tmp = t_0
	elif NdChar <= 1.65e+101:
		tmp = NaChar / (math.exp((Vef / KbT)) - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) - -1.0))
	tmp = 0.0
	if (NdChar <= -9.8e-58)
		tmp = t_0;
	elseif (NdChar <= 1.65e+101)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) - -1.0);
	tmp = 0.0;
	if (NdChar <= -9.8e-58)
		tmp = t_0;
	elseif (NdChar <= 1.65e+101)
		tmp = NaChar / (exp((Vef / KbT)) - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -9.8e-58], t$95$0, If[LessEqual[NdChar, 1.65e+101], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -9.80000000000000061e-58 or 1.65000000000000006e101 < NdChar

   1. Initial program 99.8%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in mu around 0

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

   8. Applied rewrites 66.0%

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

   if -9.80000000000000061e-58 < NdChar < 1.65000000000000006e101

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in NaChar around inf

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in Vef around inf

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

   10. Applied rewrites 50.7%

       \[\leadsto \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -9.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} - -1}\\ \mathbf{elif}\;NdChar \leq 1.65 \cdot 10^{+101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{if}\;NdChar \leq -1.95 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 7.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- (exp (/ EDonor KbT)) -1.0))))
   (if (<= NdChar -1.95e+124)
     t_0
     (if (<= NdChar 7.8e+139) (/ NaChar (- (exp (/ Vef KbT)) -1.0)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((EDonor / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -1.95e+124) {
		tmp = t_0;
	} else if (NdChar <= 7.8e+139) {
		tmp = NaChar / (exp((Vef / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp((edonor / kbt)) - (-1.0d0))
    if (ndchar <= (-1.95d+124)) then
        tmp = t_0
    else if (ndchar <= 7.8d+139) then
        tmp = nachar / (exp((vef / kbt)) - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((EDonor / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -1.95e+124) {
		tmp = t_0;
	} else if (NdChar <= 7.8e+139) {
		tmp = NaChar / (Math.exp((Vef / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((EDonor / KbT)) - -1.0)
	tmp = 0
	if NdChar <= -1.95e+124:
		tmp = t_0
	elif NdChar <= 7.8e+139:
		tmp = NaChar / (math.exp((Vef / KbT)) - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) - -1.0))
	tmp = 0.0
	if (NdChar <= -1.95e+124)
		tmp = t_0;
	elseif (NdChar <= 7.8e+139)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((EDonor / KbT)) - -1.0);
	tmp = 0.0;
	if (NdChar <= -1.95e+124)
		tmp = t_0;
	elseif (NdChar <= 7.8e+139)
		tmp = NaChar / (exp((Vef / 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[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.95e+124], t$95$0, If[LessEqual[NdChar, 7.8e+139], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.95e124 or 7.80000000000000012e139 < NdChar

   1. Initial program 99.7%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 56.2%

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

   if -1.95e124 < NdChar < 7.80000000000000012e139

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in NaChar around inf

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 70.5

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in Vef around inf

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

   10. Applied rewrites 48.0%

       \[\leadsto \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.95 \cdot 10^{+124}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{elif}\;NdChar \leq 7.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 40.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{if}\;NdChar \leq -1.8 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 1.56 \cdot 10^{+140}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- (exp (/ EDonor KbT)) -1.0))))
   (if (<= NdChar -1.8e+47)
     t_0
     (if (<= NdChar 1.56e+140) (/ NaChar (- (exp (/ Ev KbT)) -1.0)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((EDonor / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -1.8e+47) {
		tmp = t_0;
	} else if (NdChar <= 1.56e+140) {
		tmp = NaChar / (exp((Ev / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp((edonor / kbt)) - (-1.0d0))
    if (ndchar <= (-1.8d+47)) then
        tmp = t_0
    else if (ndchar <= 1.56d+140) then
        tmp = nachar / (exp((ev / kbt)) - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((EDonor / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -1.8e+47) {
		tmp = t_0;
	} else if (NdChar <= 1.56e+140) {
		tmp = NaChar / (Math.exp((Ev / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((EDonor / KbT)) - -1.0)
	tmp = 0
	if NdChar <= -1.8e+47:
		tmp = t_0
	elif NdChar <= 1.56e+140:
		tmp = NaChar / (math.exp((Ev / KbT)) - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) - -1.0))
	tmp = 0.0
	if (NdChar <= -1.8e+47)
		tmp = t_0;
	elseif (NdChar <= 1.56e+140)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((EDonor / KbT)) - -1.0);
	tmp = 0.0;
	if (NdChar <= -1.8e+47)
		tmp = t_0;
	elseif (NdChar <= 1.56e+140)
		tmp = NaChar / (exp((Ev / 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[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.8e+47], t$95$0, If[LessEqual[NdChar, 1.56e+140], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.80000000000000004e47 or 1.56000000000000002e140 < NdChar

   1. Initial program 99.8%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 54.1%

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

   if -1.80000000000000004e47 < NdChar < 1.56000000000000002e140

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in NaChar around inf

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 71.7

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in Ev around inf

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

   10. Applied rewrites 45.4%

       \[\leadsto \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{elif}\;NdChar \leq 1.56 \cdot 10^{+140}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 41.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{if}\;NdChar \leq -2.3 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 9 \cdot 10^{+139}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- (exp (/ EDonor KbT)) -1.0))))
   (if (<= NdChar -2.3e+47)
     t_0
     (if (<= NdChar 9e+139) (/ NaChar (- (exp (/ EAccept KbT)) -1.0)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((EDonor / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -2.3e+47) {
		tmp = t_0;
	} else if (NdChar <= 9e+139) {
		tmp = NaChar / (exp((EAccept / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp((edonor / kbt)) - (-1.0d0))
    if (ndchar <= (-2.3d+47)) then
        tmp = t_0
    else if (ndchar <= 9d+139) then
        tmp = nachar / (exp((eaccept / kbt)) - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp((EDonor / KbT)) - -1.0);
	double tmp;
	if (NdChar <= -2.3e+47) {
		tmp = t_0;
	} else if (NdChar <= 9e+139) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((EDonor / KbT)) - -1.0)
	tmp = 0
	if NdChar <= -2.3e+47:
		tmp = t_0
	elif NdChar <= 9e+139:
		tmp = NaChar / (math.exp((EAccept / KbT)) - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) - -1.0))
	tmp = 0.0
	if (NdChar <= -2.3e+47)
		tmp = t_0;
	elseif (NdChar <= 9e+139)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((EDonor / KbT)) - -1.0);
	tmp = 0.0;
	if (NdChar <= -2.3e+47)
		tmp = t_0;
	elseif (NdChar <= 9e+139)
		tmp = NaChar / (exp((EAccept / 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[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -2.3e+47], t$95$0, If[LessEqual[NdChar, 9e+139], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.2999999999999999e47 or 8.9999999999999999e139 < NdChar

   1. Initial program 99.8%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 54.1%

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

   if -2.2999999999999999e47 < NdChar < 8.9999999999999999e139

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in NaChar around inf

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 71.7

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in EAccept around inf

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

   10. Applied rewrites 43.1%

       \[\leadsto \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{elif}\;NdChar \leq 9 \cdot 10^{+139}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 40.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{if}\;KbT \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{NaChar}{KbT} \cdot Ev, t\_0\right)\\ \mathbf{elif}\;KbT \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{-NdChar}{KbT} \cdot Ec, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5)))
   (if (<= KbT -5.2e+131)
     (fma -0.25 (* (/ NaChar KbT) Ev) t_0)
     (if (<= KbT 3.5e+69)
       (/ NdChar (- (exp (/ EDonor KbT)) -1.0))
       (fma -0.25 (* (/ (- NdChar) KbT) Ec) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double tmp;
	if (KbT <= -5.2e+131) {
		tmp = fma(-0.25, ((NaChar / KbT) * Ev), t_0);
	} else if (KbT <= 3.5e+69) {
		tmp = NdChar / (exp((EDonor / KbT)) - -1.0);
	} else {
		tmp = fma(-0.25, ((-NdChar / KbT) * Ec), t_0);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	tmp = 0.0
	if (KbT <= -5.2e+131)
		tmp = fma(-0.25, Float64(Float64(NaChar / KbT) * Ev), t_0);
	elseif (KbT <= 3.5e+69)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) - -1.0));
	else
		tmp = fma(-0.25, Float64(Float64(Float64(-NdChar) / KbT) * Ec), t_0);
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[KbT, -5.2e+131], N[(-0.25 * N[(N[(NaChar / KbT), $MachinePrecision] * Ev), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[KbT, 3.5e+69], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[((-NdChar) / KbT), $MachinePrecision] * Ec), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -5.2e131

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-neg-in N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in Ev around inf

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

   8. Applied rewrites 61.5%

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

   if -5.2e131 < KbT < 3.49999999999999987e69

   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 NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 37.1%

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

   if 3.49999999999999987e69 < KbT

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. distribute-neg-in N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in Ec around inf

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

   8. Applied rewrites 52.1%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{NaChar}{KbT} \cdot Ev, \left(NaChar + NdChar\right) \cdot 0.5\right)\\ \mathbf{elif}\;KbT \leq 3.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{-NdChar}{KbT} \cdot Ec, \left(NaChar + NdChar\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 22.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{elif}\;NdChar \leq 1.56 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -2e+47)
   (* 0.5 NdChar)
   (if (<= NdChar 1.56e+140) (* 0.5 NaChar) (* 0.5 NdChar))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -2e+47) {
		tmp = 0.5 * NdChar;
	} else if (NdChar <= 1.56e+140) {
		tmp = 0.5 * NaChar;
	} else {
		tmp = 0.5 * NdChar;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-2d+47)) then
        tmp = 0.5d0 * ndchar
    else if (ndchar <= 1.56d+140) then
        tmp = 0.5d0 * nachar
    else
        tmp = 0.5d0 * ndchar
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -2e+47) {
		tmp = 0.5 * NdChar;
	} else if (NdChar <= 1.56e+140) {
		tmp = 0.5 * NaChar;
	} else {
		tmp = 0.5 * NdChar;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -2e+47:
		tmp = 0.5 * NdChar
	elif NdChar <= 1.56e+140:
		tmp = 0.5 * NaChar
	else:
		tmp = 0.5 * NdChar
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -2e+47)
		tmp = Float64(0.5 * NdChar);
	elseif (NdChar <= 1.56e+140)
		tmp = Float64(0.5 * NaChar);
	else
		tmp = Float64(0.5 * NdChar);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -2e+47)
		tmp = 0.5 * NdChar;
	elseif (NdChar <= 1.56e+140)
		tmp = 0.5 * NaChar;
	else
		tmp = 0.5 * NdChar;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -2e+47], N[(0.5 * NdChar), $MachinePrecision], If[LessEqual[NdChar, 1.56e+140], N[(0.5 * NaChar), $MachinePrecision], N[(0.5 * NdChar), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.0000000000000001e47 or 1.56000000000000002e140 < NdChar

   1. Initial program 99.8%

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

   3. Taylor expanded in NaChar around 0

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 26.8%

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

   if -2.0000000000000001e47 < NdChar < 1.56000000000000002e140

   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 28.0

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

   5. Applied rewrites 28.0%

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

   6. Taylor expanded in NaChar around inf

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

   8. Applied rewrites 24.7%

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

Alternative 24: 28.2% accurate, 30.7× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 28.8

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

5. Applied rewrites 28.8%

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

6. Final simplification 28.8%

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

Alternative 25: 18.8% accurate, 46.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 28.8

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

5. Applied rewrites 28.8%

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

6. Taylor expanded in NaChar around inf

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

8. Applied rewrites 20.0%

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

Reproduce

herbie shell --seed 2024249 
(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))))))