Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 15.9s

Alternatives: 24

Speedup: 1.0×

• could not determine a ground truth

  1. branch1543363 = 6.720934637617518e-116
  2. NdChar = 3.6704797913141255e-116
  3. Ec = -1.8130289377781448e-300
  4. Vef = -1.4665132904181677e+268
  5. EDonor = 5.818823890659561e+64
  6. mu = 1.3290785267463158e-145
  7. KbT = 1.0400503294052072e-18
  8. NaChar = -6.649417828067031e+21
  9. Ev = -1.7304094649309288e-183
  10. EAccept = 468920272630241.94

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\mathsf{E}\left(\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT} \cdot \left(-0.5\right)\right)}\\ \frac{NdChar}{\mathsf{fma}\left(t\_0, t\_0, 1\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (pow (E) (* (/ (- (- (- Ec Vef) EDonor) mu) KbT) (- 0.5)))))
   (+
    (/ NdChar (fma t_0 t_0 1.0))
    (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. sqr-pow N/A

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

   5. lower-fma.f64 N/A

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \frac{NdChar}{\mathsf{fma}\left({\mathsf{E}\left(\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT} \cdot \left(-0.5\right)\right)}, {\mathsf{E}\left(\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT} \cdot \left(-0.5\right)\right)}, 1\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \]
8. Add Preprocessing

Alternative 2: 77.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_2 := t\_0 + \frac{NaChar}{1 + t\_1}\\ \mathbf{if}\;t\_2 \leq -8 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-205} \lor \neg \left(t\_2 \leq 10^{-95}\right):\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_1 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_2 (+ t_0 (/ NaChar (+ 1.0 t_1)))))
   (if (<= t_2 -8e+21)
     (+
      (/ NdChar (+ (exp (/ (- EDonor Ec) KbT)) 1.0))
      (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))
     (if (or (<= t_2 -2e-205) (not (<= t_2 1e-95)))
       (+ t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
       (/ NaChar (+ t_1 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 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = t_0 + (NaChar / (1.0 + t_1));
	double tmp;
	if (t_2 <= -8e+21) {
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if ((t_2 <= -2e-205) || !(t_2 <= 1e-95)) {
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	} else {
		tmp = NaChar / (t_1 + 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_1 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_2 = t_0 + (nachar / (1.0d0 + t_1))
    if (t_2 <= (-8d+21)) then
        tmp = (ndchar / (exp(((edonor - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    else if ((t_2 <= (-2d-205)) .or. (.not. (t_2 <= 1d-95))) then
        tmp = t_0 + (nachar / (1.0d0 + exp((ev / kbt))))
    else
        tmp = nachar / (t_1 + 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 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = t_0 + (NaChar / (1.0 + t_1));
	double tmp;
	if (t_2 <= -8e+21) {
		tmp = (NdChar / (Math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if ((t_2 <= -2e-205) || !(t_2 <= 1e-95)) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else {
		tmp = NaChar / (t_1 + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = t_0 + (NaChar / (1.0 + t_1))
	tmp = 0
	if t_2 <= -8e+21:
		tmp = (NdChar / (math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	elif (t_2 <= -2e-205) or not (t_2 <= 1e-95):
		tmp = t_0 + (NaChar / (1.0 + math.exp((Ev / KbT))))
	else:
		tmp = NaChar / (t_1 + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_1 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = Float64(t_0 + Float64(NaChar / Float64(1.0 + t_1)))
	tmp = 0.0
	if (t_2 <= -8e+21)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	elseif ((t_2 <= -2e-205) || !(t_2 <= 1e-95))
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	else
		tmp = Float64(NaChar / Float64(t_1 + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_2 = t_0 + (NaChar / (1.0 + t_1));
	tmp = 0.0;
	if (t_2 <= -8e+21)
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	elseif ((t_2 <= -2e-205) || ~((t_2 <= 1e-95)))
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	else
		tmp = NaChar / (t_1 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -8e+21], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, -2e-205], N[Not[LessEqual[t$95$2, 1e-95]], $MachinePrecision]], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -8e21

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in mu around 0

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

   8. Applied rewrites 86.3%

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

   if -8e21 < (+.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-205 or 9.99999999999999989e-96 < (+.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 Ev around inf

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

      1. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -2e-205 < (+.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))))) < 9.99999999999999989e-96

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -8 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-205} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 10^{-95}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_2 := \frac{NaChar}{1 + t\_1}\\ t_3 := 0.5 \cdot NdChar + t\_2\\ t_4 := t\_0 + t\_2\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{-198}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{NaChar}{t\_1 + 1}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+115}:\\ \;\;\;\;t\_0 + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_2 (/ NaChar (+ 1.0 t_1)))
        (t_3 (+ (* 0.5 NdChar) t_2))
        (t_4 (+ t_0 t_2)))
   (if (<= t_4 -5e-198)
     t_3
     (if (<= t_4 2e-54)
       (/ NaChar (+ t_1 1.0))
       (if (<= t_4 5e+115) (+ t_0 (/ NaChar 2.0)) t_3)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = NaChar / (1.0 + t_1);
	double t_3 = (0.5 * NdChar) + t_2;
	double t_4 = t_0 + t_2;
	double tmp;
	if (t_4 <= -5e-198) {
		tmp = t_3;
	} else if (t_4 <= 2e-54) {
		tmp = NaChar / (t_1 + 1.0);
	} else if (t_4 <= 5e+115) {
		tmp = t_0 + (NaChar / 2.0);
	} else {
		tmp = t_3;
	}
	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) :: t_4
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_1 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_2 = nachar / (1.0d0 + t_1)
    t_3 = (0.5d0 * ndchar) + t_2
    t_4 = t_0 + t_2
    if (t_4 <= (-5d-198)) then
        tmp = t_3
    else if (t_4 <= 2d-54) then
        tmp = nachar / (t_1 + 1.0d0)
    else if (t_4 <= 5d+115) then
        tmp = t_0 + (nachar / 2.0d0)
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = NaChar / (1.0 + t_1);
	double t_3 = (0.5 * NdChar) + t_2;
	double t_4 = t_0 + t_2;
	double tmp;
	if (t_4 <= -5e-198) {
		tmp = t_3;
	} else if (t_4 <= 2e-54) {
		tmp = NaChar / (t_1 + 1.0);
	} else if (t_4 <= 5e+115) {
		tmp = t_0 + (NaChar / 2.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = NaChar / (1.0 + t_1)
	t_3 = (0.5 * NdChar) + t_2
	t_4 = t_0 + t_2
	tmp = 0
	if t_4 <= -5e-198:
		tmp = t_3
	elif t_4 <= 2e-54:
		tmp = NaChar / (t_1 + 1.0)
	elif t_4 <= 5e+115:
		tmp = t_0 + (NaChar / 2.0)
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_1 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = Float64(NaChar / Float64(1.0 + t_1))
	t_3 = Float64(Float64(0.5 * NdChar) + t_2)
	t_4 = Float64(t_0 + t_2)
	tmp = 0.0
	if (t_4 <= -5e-198)
		tmp = t_3;
	elseif (t_4 <= 2e-54)
		tmp = Float64(NaChar / Float64(t_1 + 1.0));
	elseif (t_4 <= 5e+115)
		tmp = Float64(t_0 + Float64(NaChar / 2.0));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_2 = NaChar / (1.0 + t_1);
	t_3 = (0.5 * NdChar) + t_2;
	t_4 = t_0 + t_2;
	tmp = 0.0;
	if (t_4 <= -5e-198)
		tmp = t_3;
	elseif (t_4 <= 2e-54)
		tmp = NaChar / (t_1 + 1.0);
	elseif (t_4 <= 5e+115)
		tmp = t_0 + (NaChar / 2.0);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, -5e-198], t$95$3, If[LessEqual[t$95$4, 2e-54], N[(NaChar / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+115], N[(t$95$0 + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

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))))) < -4.9999999999999999e-198 or 5.00000000000000008e115 < (+.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

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

      1. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -4.9999999999999999e-198 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.0000000000000001e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if 2.0000000000000001e-54 < (+.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))))) < 5.00000000000000008e115

   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

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

   5. Applied rewrites 72.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_1 := \frac{NaChar}{1 + t\_0}\\ t_2 := 0.5 \cdot NdChar + t\_1\\ t_3 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + t\_1\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-198}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-28}:\\ \;\;\;\;\frac{NaChar}{t\_0 + 1}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_1 (/ NaChar (+ 1.0 t_0)))
        (t_2 (+ (* 0.5 NdChar) t_1))
        (t_3
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          t_1)))
   (if (<= t_3 -5e-198)
     t_2
     (if (<= t_3 5e-28)
       (/ NaChar (+ t_0 1.0))
       (if (<= t_3 5e+115)
         (/ NdChar (+ (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) 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 = NaChar / (1.0 + t_0);
	double t_2 = (0.5 * NdChar) + t_1;
	double t_3 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_1;
	double tmp;
	if (t_3 <= -5e-198) {
		tmp = t_2;
	} else if (t_3 <= 5e-28) {
		tmp = NaChar / (t_0 + 1.0);
	} else if (t_3 <= 5e+115) {
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 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 = nachar / (1.0d0 + t_0)
    t_2 = (0.5d0 * ndchar) + t_1
    t_3 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + t_1
    if (t_3 <= (-5d-198)) then
        tmp = t_2
    else if (t_3 <= 5d-28) then
        tmp = nachar / (t_0 + 1.0d0)
    else if (t_3 <= 5d+115) then
        tmp = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) + 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 = NaChar / (1.0 + t_0);
	double t_2 = (0.5 * NdChar) + t_1;
	double t_3 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_1;
	double tmp;
	if (t_3 <= -5e-198) {
		tmp = t_2;
	} else if (t_3 <= 5e-28) {
		tmp = NaChar / (t_0 + 1.0);
	} else if (t_3 <= 5e+115) {
		tmp = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 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 = NaChar / (1.0 + t_0)
	t_2 = (0.5 * NdChar) + t_1
	t_3 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_1
	tmp = 0
	if t_3 <= -5e-198:
		tmp = t_2
	elif t_3 <= 5e-28:
		tmp = NaChar / (t_0 + 1.0)
	elif t_3 <= 5e+115:
		tmp = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 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 = Float64(NaChar / Float64(1.0 + t_0))
	t_2 = Float64(Float64(0.5 * NdChar) + t_1)
	t_3 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + t_1)
	tmp = 0.0
	if (t_3 <= -5e-198)
		tmp = t_2;
	elseif (t_3 <= 5e-28)
		tmp = Float64(NaChar / Float64(t_0 + 1.0));
	elseif (t_3 <= 5e+115)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) + 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 = NaChar / (1.0 + t_0);
	t_2 = (0.5 * NdChar) + t_1;
	t_3 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_1;
	tmp = 0.0;
	if (t_3 <= -5e-198)
		tmp = t_2;
	elseif (t_3 <= 5e-28)
		tmp = NaChar / (t_0 + 1.0);
	elseif (t_3 <= 5e+115)
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 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[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-198], t$95$2, If[LessEqual[t$95$3, 5e-28], N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+115], N[(NdChar / N[(N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.9999999999999999e-198 or 5.00000000000000008e115 < (+.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

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

      1. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -4.9999999999999999e-198 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5.0000000000000002e-28

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if 5.0000000000000002e-28 < (+.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))))) < 5.00000000000000008e115

   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. 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. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. 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 74.9

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

   7. Applied rewrites 74.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-28}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-205}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NaChar}{2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \frac{Vef - mu}{KbT}\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+179}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (<= t_1 -2e-205)
     t_0
     (if (<= t_1 0.0)
       (/ NaChar (+ 2.0 (+ (/ EAccept KbT) (+ (/ Ev KbT) (/ (- Vef mu) KbT)))))
       (if (<= t_1 1e+179) (/ NdChar (+ (exp (/ mu KbT)) 1.0)) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-205) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = NaChar / (2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef - mu) / KbT))));
	} else if (t_1 <= 1e+179) {
		tmp = NdChar / (exp((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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if (t_1 <= (-2d-205)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = nachar / (2.0d0 + ((eaccept / kbt) + ((ev / kbt) + ((vef - mu) / kbt))))
    else if (t_1 <= 1d+179) then
        tmp = ndchar / (exp((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 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-205) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = NaChar / (2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef - mu) / KbT))));
	} else if (t_1 <= 1e+179) {
		tmp = NdChar / (Math.exp((mu / 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 * (NdChar + NaChar)
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if t_1 <= -2e-205:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = NaChar / (2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef - mu) / KbT))))
	elif t_1 <= 1e+179:
		tmp = NdChar / (math.exp((mu / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = 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) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-205)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Ev / KbT) + Float64(Float64(Vef - mu) / KbT)))));
	elseif (t_1 <= 1e+179)
		tmp = Float64(NdChar / Float64(exp(Float64(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 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -2e-205)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = NaChar / (2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef - mu) / KbT))));
	elseif (t_1 <= 1e+179)
		tmp = NdChar / (exp((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[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -2e-205], t$95$0, If[LessEqual[t$95$1, 0.0], N[(NaChar / N[(2.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(N[(Vef - mu), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+179], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-205 or 9.9999999999999998e178 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 39.9

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

   5. Applied rewrites 39.9%

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

   if -2e-205 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in Ev around 0

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

   8. Applied rewrites 95.6%

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

   9. Taylor expanded in KbT around inf

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

   11. Applied rewrites 56.8%

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

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999998e178

   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 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 NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. 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 54.9

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

   7. Applied rewrites 54.9%

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

   8. Taylor expanded in mu around inf

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

   10. Applied rewrites 42.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-205}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{NaChar}{2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \frac{Vef - mu}{KbT}\right)\right)}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 10^{+179}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-205} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 t_0)))))
   (if (or (<= t_1 -2e-205) (not (<= t_1 0.0)))
     (+
      (/ NdChar (+ (exp (/ (- (+ mu EDonor) Ec) KbT)) 1.0))
      (/ NaChar (+ (exp (/ (- (+ EAccept Ev) mu) KbT)) 1.0)))
     (/ NaChar (+ t_0 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	double tmp;
	if ((t_1 <= -2e-205) || !(t_1 <= 0.0)) {
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	} else {
		tmp = NaChar / (t_0 + 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) :: t_1
    real(8) :: tmp
    t_0 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + t_0))
    if ((t_1 <= (-2d-205)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = (ndchar / (exp((((mu + edonor) - ec) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + ev) - mu) / kbt)) + 1.0d0))
    else
        tmp = nachar / (t_0 + 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 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	double tmp;
	if ((t_1 <= -2e-205) || !(t_1 <= 0.0)) {
		tmp = (NdChar / (Math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	} else {
		tmp = NaChar / (t_0 + 1.0);
	}
	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 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0))
	tmp = 0
	if (t_1 <= -2e-205) or not (t_1 <= 0.0):
		tmp = (NdChar / (math.exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + Ev) - mu) / KbT)) + 1.0))
	else:
		tmp = NaChar / (t_0 + 1.0)
	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 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + t_0)))
	tmp = 0.0
	if ((t_1 <= -2e-205) || !(t_1 <= 0.0))
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + EDonor) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)) + 1.0)));
	else
		tmp = Float64(NaChar / Float64(t_0 + 1.0));
	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 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	tmp = 0.0;
	if ((t_1 <= -2e-205) || ~((t_1 <= 0.0)))
		tmp = (NdChar / (exp((((mu + EDonor) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + Ev) - mu) / KbT)) + 1.0));
	else
		tmp = NaChar / (t_0 + 1.0);
	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[(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 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-205], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

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-205 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 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   5. Applied rewrites 90.8%

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

   if -2e-205 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-205} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 0\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_2 := t\_0 + \frac{NaChar}{1 + t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-205}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\frac{NaChar}{t\_1 + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_2 (+ t_0 (/ NaChar (+ 1.0 t_1)))))
   (if (<= t_2 -2e-205)
     (+
      (/ NdChar (+ (exp (/ (- EDonor Ec) KbT)) 1.0))
      (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))
     (if (<= t_2 4e-101)
       (/ NaChar (+ t_1 1.0))
       (+ t_0 (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = t_0 + (NaChar / (1.0 + t_1));
	double tmp;
	if (t_2 <= -2e-205) {
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (t_2 <= 4e-101) {
		tmp = NaChar / (t_1 + 1.0);
	} else {
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_1 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_2 = t_0 + (nachar / (1.0d0 + t_1))
    if (t_2 <= (-2d-205)) then
        tmp = (ndchar / (exp(((edonor - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    else if (t_2 <= 4d-101) then
        tmp = nachar / (t_1 + 1.0d0)
    else
        tmp = t_0 + (nachar / (1.0d0 + exp((eaccept / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_2 = t_0 + (NaChar / (1.0 + t_1));
	double tmp;
	if (t_2 <= -2e-205) {
		tmp = (NdChar / (Math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	} else if (t_2 <= 4e-101) {
		tmp = NaChar / (t_1 + 1.0);
	} else {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = t_0 + (NaChar / (1.0 + t_1))
	tmp = 0
	if t_2 <= -2e-205:
		tmp = (NdChar / (math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	elif t_2 <= 4e-101:
		tmp = NaChar / (t_1 + 1.0)
	else:
		tmp = t_0 + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_1 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_2 = Float64(t_0 + Float64(NaChar / Float64(1.0 + t_1)))
	tmp = 0.0
	if (t_2 <= -2e-205)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	elseif (t_2 <= 4e-101)
		tmp = Float64(NaChar / Float64(t_1 + 1.0));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_2 = t_0 + (NaChar / (1.0 + t_1));
	tmp = 0.0;
	if (t_2 <= -2e-205)
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	elseif (t_2 <= 4e-101)
		tmp = NaChar / (t_1 + 1.0);
	else
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-205], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-101], N[(NaChar / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-205

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in mu around 0

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

   8. Applied rewrites 80.0%

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

   if -2e-205 < (+.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))))) < 4.00000000000000021e-101

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 4.00000000000000021e-101 < (+.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 EAccept around inf

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

      1. lower-/.f64 78.8

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

   5. Applied rewrites 78.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-205}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-205} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 t_0)))))
   (if (or (<= t_1 -2e-205) (not (<= t_1 2e-28)))
     (+
      (/ NdChar (+ (exp (/ (- EDonor Ec) KbT)) 1.0))
      (/ NaChar (+ (exp (/ (+ EAccept Ev) KbT)) 1.0)))
     (/ NaChar (+ t_0 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	double tmp;
	if ((t_1 <= -2e-205) || !(t_1 <= 2e-28)) {
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	} else {
		tmp = NaChar / (t_0 + 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) :: t_1
    real(8) :: tmp
    t_0 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + t_0))
    if ((t_1 <= (-2d-205)) .or. (.not. (t_1 <= 2d-28))) then
        tmp = (ndchar / (exp(((edonor - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((eaccept + ev) / kbt)) + 1.0d0))
    else
        tmp = nachar / (t_0 + 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 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	double tmp;
	if ((t_1 <= -2e-205) || !(t_1 <= 2e-28)) {
		tmp = (NdChar / (Math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((EAccept + Ev) / KbT)) + 1.0));
	} else {
		tmp = NaChar / (t_0 + 1.0);
	}
	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 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0))
	tmp = 0
	if (t_1 <= -2e-205) or not (t_1 <= 2e-28):
		tmp = (NdChar / (math.exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((EAccept + Ev) / KbT)) + 1.0))
	else:
		tmp = NaChar / (t_0 + 1.0)
	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 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + t_0)))
	tmp = 0.0
	if ((t_1 <= -2e-205) || !(t_1 <= 2e-28))
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(EDonor - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Ev) / KbT)) + 1.0)));
	else
		tmp = Float64(NaChar / Float64(t_0 + 1.0));
	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 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	tmp = 0.0;
	if ((t_1 <= -2e-205) || ~((t_1 <= 2e-28)))
		tmp = (NdChar / (exp(((EDonor - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((EAccept + Ev) / KbT)) + 1.0));
	else
		tmp = NaChar / (t_0 + 1.0);
	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[(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 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-205], N[Not[LessEqual[t$95$1, 2e-28]], $MachinePrecision]], N[(N[(NdChar / N[(N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

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-205 or 1.99999999999999994e-28 < (+.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 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in mu around 0

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

   8. Applied rewrites 79.5%

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

   if -2e-205 < (+.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))))) < 1.99999999999999994e-28

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-205} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{EAccept + Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-266} \lor \neg \left(t\_0 \leq 4 \cdot 10^{-187}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(NdChar \cdot NdChar\right) \cdot {\left(NdChar - NaChar\right)}^{-1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-266) (not (<= t_0 4e-187)))
     (* 0.5 (+ NdChar NaChar))
     (* 0.5 (* (* NdChar NdChar) (pow (- NdChar NaChar) -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 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-266) || !(t_0 <= 4e-187)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * ((NdChar * NdChar) * pow((NdChar - NaChar), -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 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-2d-266)) .or. (.not. (t_0 <= 4d-187))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = 0.5d0 * ((ndchar * ndchar) * ((ndchar - nachar) ** (-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 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-266) || !(t_0 <= 4e-187)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * ((NdChar * NdChar) * Math.pow((NdChar - NaChar), -1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -2e-266) or not (t_0 <= 4e-187):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = 0.5 * ((NdChar * NdChar) * math.pow((NdChar - NaChar), -1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 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) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -2e-266) || !(t_0 <= 4e-187))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(0.5 * Float64(Float64(NdChar * NdChar) * (Float64(NdChar - NaChar) ^ -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -2e-266) || ~((t_0 <= 4e-187)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = 0.5 * ((NdChar * NdChar) * ((NdChar - NaChar) ^ -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -2e-266], N[Not[LessEqual[t$95$0, 4e-187]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(NdChar * NdChar), $MachinePrecision] * N[Power[N[(NdChar - NaChar), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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-266 or 4.0000000000000001e-187 < (+.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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 35.7

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

   5. Applied rewrites 35.7%

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

   if -2e-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))))) < 4.0000000000000001e-187

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in NdChar around inf

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

   10. Applied rewrites 27.3%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-266} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 4 \cdot 10^{-187}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(NdChar \cdot NdChar\right) \cdot {\left(NdChar - NaChar\right)}^{-1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-205} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-111}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \frac{Vef - mu}{KbT}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-205) (not (<= t_0 2e-111)))
     (* 0.5 (+ NdChar NaChar))
     (/
      NaChar
      (+ 2.0 (+ (/ EAccept KbT) (+ (/ Ev KbT) (/ (- Vef mu) KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-205) || !(t_0 <= 2e-111)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NaChar / (2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef - mu) / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-2d-205)) .or. (.not. (t_0 <= 2d-111))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = nachar / (2.0d0 + ((eaccept / kbt) + ((ev / kbt) + ((vef - mu) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-205) || !(t_0 <= 2e-111)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NaChar / (2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef - mu) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -2e-205) or not (t_0 <= 2e-111):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = NaChar / (2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef - mu) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 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) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -2e-205) || !(t_0 <= 2e-111))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Ev / KbT) + Float64(Float64(Vef - mu) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -2e-205) || ~((t_0 <= 2e-111)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = NaChar / (2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef - mu) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -2e-205], N[Not[LessEqual[t$95$0, 2e-111]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(2.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(N[(Vef - mu), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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-205 or 2.00000000000000018e-111 < (+.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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 37.0

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

   5. Applied rewrites 37.0%

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

   if -2e-205 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.00000000000000018e-111

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in Ev around 0

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

   8. Applied rewrites 87.5%

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

   9. Taylor expanded in KbT around inf

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

   11. Applied rewrites 48.5%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-205} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-111}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \frac{Vef - mu}{KbT}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-205} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-176}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{NdChar - NaChar} \cdot \left(-0.25 \cdot \left(NaChar \cdot NaChar\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-205) (not (<= t_0 5e-176)))
     (* 0.5 (+ NdChar NaChar))
     (* (/ 2.0 (- NdChar NaChar)) (* -0.25 (* NaChar 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 / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-205) || !(t_0 <= 5e-176)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = (2.0 / (NdChar - NaChar)) * (-0.25 * (NaChar * 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 / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-2d-205)) .or. (.not. (t_0 <= 5d-176))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = (2.0d0 / (ndchar - nachar)) * ((-0.25d0) * (nachar * 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 / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-205) || !(t_0 <= 5e-176)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = (2.0 / (NdChar - NaChar)) * (-0.25 * (NaChar * NaChar));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -2e-205) or not (t_0 <= 5e-176):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = (2.0 / (NdChar - NaChar)) * (-0.25 * (NaChar * NaChar))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 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) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -2e-205) || !(t_0 <= 5e-176))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(Float64(2.0 / Float64(NdChar - NaChar)) * Float64(-0.25 * Float64(NaChar * NaChar)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -2e-205) || ~((t_0 <= 5e-176)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = (2.0 / (NdChar - NaChar)) * (-0.25 * (NaChar * 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[(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]}, If[Or[LessEqual[t$95$0, -2e-205], N[Not[LessEqual[t$95$0, 5e-176]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(NdChar - NaChar), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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-205 or 5e-176 < (+.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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 36.4

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

   5. Applied rewrites 36.4%

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

   if -2e-205 < (+.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))))) < 5e-176

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   7. Applied rewrites 10.2%

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

   9. Applied rewrites 10.2%

      \[\leadsto \frac{2}{NdChar - NaChar} \cdot \color{blue}{\left(0.25 \cdot \left(\left(NdChar - NaChar\right) \cdot \left(NaChar + NdChar\right)\right)\right)} \]

   10. Taylor expanded in NdChar around 0

       \[\leadsto \frac{2}{NdChar - NaChar} \cdot \left(\frac{-1}{4} \cdot \color{blue}{{NaChar}^{2}}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 26.6%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-205} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-176}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{NdChar - NaChar} \cdot \left(-0.25 \cdot \left(NaChar \cdot NaChar\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -7 \cdot 10^{+100} \lor \neg \left(Vef \leq 3.8 \cdot 10^{+122}\right):\\ \;\;\;\;\frac{NdChar}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= Vef -7e+100) (not (<= Vef 3.8e+122)))
   (+
    (/ NdChar (+ 1.0 (pow (E) (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
    (/ NaChar (+ 1.0 (exp (/ Vef KbT)))))
   (+
    (/ NdChar (+ (exp (/ (- (+ mu EDonor) Ec) KbT)) 1.0))
    (/ NaChar (+ (exp (/ (- (+ EAccept Ev) mu) KbT)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if Vef < -6.99999999999999953e100 or 3.7999999999999998e122 < Vef

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-1-e N/A

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

      3. lower-E.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in Vef around inf

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

      1. lower-/.f64 88.2

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

   9. Applied rewrites 88.2%

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

   if -6.99999999999999953e100 < Vef < 3.7999999999999998e122

   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 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -7 \cdot 10^{+100} \lor \neg \left(Vef \leq 3.8 \cdot 10^{+122}\right):\\ \;\;\;\;\frac{NdChar}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 14: 100.0% accurate, 1.0× speedup

Math

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

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) - 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]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 15: 69.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -3.5 \cdot 10^{+79} \lor \neg \left(NaChar \leq 2.2 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -3.5e+79) (not (<= NaChar 2.2e-147)))
   (/ NaChar (+ (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) 1.0))
   (/ NdChar (+ (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -3.5e+79) || !(NaChar <= 2.2e-147)) {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((nachar <= (-3.5d+79)) .or. (.not. (nachar <= 2.2d-147))) then
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) + 1.0d0)
    else
        tmp = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -3.5e+79) || !(NaChar <= 2.2e-147)) {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -3.5e+79) or not (NaChar <= 2.2e-147):
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0)
	else:
		tmp = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -3.5e+79) || !(NaChar <= 2.2e-147))
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) + 1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -3.5e+79) || ~((NaChar <= 2.2e-147)))
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	else
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -3.5e+79], N[Not[LessEqual[NaChar, 2.2e-147]], $MachinePrecision]], 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[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -3.4999999999999998e79 or 2.2000000000000001e-147 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -3.4999999999999998e79 < NaChar < 2.2000000000000001e-147

   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 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 NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. 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 74.5

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

   7. Applied rewrites 74.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -3.5 \cdot 10^{+79} \lor \neg \left(NaChar \leq 2.2 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -1.5 \lor \neg \left(NaChar \leq 2.6 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + Vef\right) - Ec}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -1.5) (not (<= NaChar 2.6e-154)))
   (/ NaChar (+ (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) 1.0))
   (/ NdChar (+ (exp (/ (- (+ mu Vef) Ec) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -1.5) || !(NaChar <= 2.6e-154)) {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (exp((((mu + Vef) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((nachar <= (-1.5d0)) .or. (.not. (nachar <= 2.6d-154))) then
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) + 1.0d0)
    else
        tmp = ndchar / (exp((((mu + vef) - ec) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -1.5) || !(NaChar <= 2.6e-154)) {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (Math.exp((((mu + Vef) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -1.5) or not (NaChar <= 2.6e-154):
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0)
	else:
		tmp = NdChar / (math.exp((((mu + Vef) - Ec) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -1.5) || !(NaChar <= 2.6e-154))
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) + 1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -1.5) || ~((NaChar <= 2.6e-154)))
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	else
		tmp = NdChar / (exp((((mu + Vef) - Ec) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -1.5], N[Not[LessEqual[NaChar, 2.6e-154]], $MachinePrecision]], 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[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.5 or 2.6e-154 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -1.5 < NaChar < 2.6e-154

   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 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 NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. 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 76.9

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in EDonor around 0

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

   10. Applied rewrites 71.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.5 \lor \neg \left(NaChar \leq 2.6 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + Vef\right) - Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -4.8 \cdot 10^{+79} \lor \neg \left(NaChar \leq 10^{-153}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Vef\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + Vef\right) - Ec}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -4.8e+79) (not (<= NaChar 1e-153)))
   (/ NaChar (+ (exp (/ (- (+ EAccept Vef) mu) KbT)) 1.0))
   (/ NdChar (+ (exp (/ (- (+ mu Vef) Ec) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -4.8e+79) || !(NaChar <= 1e-153)) {
		tmp = NaChar / (exp((((EAccept + Vef) - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (exp((((mu + Vef) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((nachar <= (-4.8d+79)) .or. (.not. (nachar <= 1d-153))) then
        tmp = nachar / (exp((((eaccept + vef) - mu) / kbt)) + 1.0d0)
    else
        tmp = ndchar / (exp((((mu + vef) - ec) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -4.8e+79) || !(NaChar <= 1e-153)) {
		tmp = NaChar / (Math.exp((((EAccept + Vef) - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (Math.exp((((mu + Vef) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -4.8e+79) or not (NaChar <= 1e-153):
		tmp = NaChar / (math.exp((((EAccept + Vef) - mu) / KbT)) + 1.0)
	else:
		tmp = NdChar / (math.exp((((mu + Vef) - Ec) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -4.8e+79) || !(NaChar <= 1e-153))
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Vef) - mu) / KbT)) + 1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -4.8e+79) || ~((NaChar <= 1e-153)))
		tmp = NaChar / (exp((((EAccept + Vef) - mu) / KbT)) + 1.0);
	else
		tmp = NdChar / (exp((((mu + Vef) - Ec) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -4.8e+79], N[Not[LessEqual[NaChar, 1e-153]], $MachinePrecision]], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -4.79999999999999971e79 or 1.00000000000000004e-153 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 78.8

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in Ev around 0

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

   8. Applied rewrites 72.8%

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

   if -4.79999999999999971e79 < NaChar < 1.00000000000000004e-153

   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 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 NdChar around inf

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

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in EDonor around 0

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

   10. Applied rewrites 69.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -4.8 \cdot 10^{+79} \lor \neg \left(NaChar \leq 10^{-153}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Vef\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(mu + Vef\right) - Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -6.6 \cdot 10^{-218} \lor \neg \left(NaChar \leq 3.7 \cdot 10^{-168}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Vef\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{-Ec}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -6.6e-218) (not (<= NaChar 3.7e-168)))
   (/ NaChar (+ (exp (/ (- (+ EAccept Vef) mu) KbT)) 1.0))
   (/ NdChar (+ (exp (/ (- Ec) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -6.6e-218) || !(NaChar <= 3.7e-168)) {
		tmp = NaChar / (exp((((EAccept + Vef) - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (exp((-Ec / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((nachar <= (-6.6d-218)) .or. (.not. (nachar <= 3.7d-168))) then
        tmp = nachar / (exp((((eaccept + vef) - mu) / kbt)) + 1.0d0)
    else
        tmp = ndchar / (exp((-ec / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -6.6e-218) || !(NaChar <= 3.7e-168)) {
		tmp = NaChar / (Math.exp((((EAccept + Vef) - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (Math.exp((-Ec / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -6.6e-218) or not (NaChar <= 3.7e-168):
		tmp = NaChar / (math.exp((((EAccept + Vef) - mu) / KbT)) + 1.0)
	else:
		tmp = NdChar / (math.exp((-Ec / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -6.6e-218) || !(NaChar <= 3.7e-168))
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Vef) - mu) / KbT)) + 1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(-Ec) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -6.6e-218) || ~((NaChar <= 3.7e-168)))
		tmp = NaChar / (exp((((EAccept + Vef) - mu) / KbT)) + 1.0);
	else
		tmp = NdChar / (exp((-Ec / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -6.6e-218], N[Not[LessEqual[NaChar, 3.7e-168]], $MachinePrecision]], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -6.60000000000000046e-218 or 3.69999999999999997e-168 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in Ev around 0

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

   8. Applied rewrites 67.2%

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

   if -6.60000000000000046e-218 < NaChar < 3.69999999999999997e-168

   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. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. 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 84.4

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in Ec around inf

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

   10. Applied rewrites 57.2%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -6.6 \cdot 10^{-218} \lor \neg \left(NaChar \leq 3.7 \cdot 10^{-168}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Vef\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{-Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq 2.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + Vef\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;EAccept \leq 1.35 \cdot 10^{+103}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Vef\right) - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept 2.4e+90)
   (/ NaChar (+ (exp (/ (- (+ Ev Vef) mu) KbT)) 1.0))
   (if (<= EAccept 1.35e+103)
     (/ NdChar (+ (exp (/ mu KbT)) 1.0))
     (/ NaChar (+ (exp (/ (- (+ EAccept Vef) mu) KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 2.4e+90) {
		tmp = NaChar / (exp((((Ev + Vef) - mu) / KbT)) + 1.0);
	} else if (EAccept <= 1.35e+103) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((((EAccept + Vef) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (eaccept <= 2.4d+90) then
        tmp = nachar / (exp((((ev + vef) - mu) / kbt)) + 1.0d0)
    else if (eaccept <= 1.35d+103) then
        tmp = ndchar / (exp((mu / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp((((eaccept + vef) - mu) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 2.4e+90) {
		tmp = NaChar / (Math.exp((((Ev + Vef) - mu) / KbT)) + 1.0);
	} else if (EAccept <= 1.35e+103) {
		tmp = NdChar / (Math.exp((mu / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp((((EAccept + Vef) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EAccept <= 2.4e+90:
		tmp = NaChar / (math.exp((((Ev + Vef) - mu) / KbT)) + 1.0)
	elif EAccept <= 1.35e+103:
		tmp = NdChar / (math.exp((mu / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp((((EAccept + Vef) - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EAccept <= 2.4e+90)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + Vef) - mu) / KbT)) + 1.0));
	elseif (EAccept <= 1.35e+103)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Vef) - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EAccept <= 2.4e+90)
		tmp = NaChar / (exp((((Ev + Vef) - mu) / KbT)) + 1.0);
	elseif (EAccept <= 1.35e+103)
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	else
		tmp = NaChar / (exp((((EAccept + Vef) - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EAccept, 2.4e+90], N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[EAccept, 1.35e+103], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if EAccept < 2.4000000000000001e90

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around 0

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

      1. 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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in EAccept around 0

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

   8. Applied rewrites 62.7%

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

   if 2.4000000000000001e90 < EAccept < 1.34999999999999996e103

   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 NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. 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 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in mu around inf

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

   10. Applied rewrites 72.9%

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

   if 1.34999999999999996e103 < EAccept

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. 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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in Ev around 0

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

   8. Applied rewrites 73.1%

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

Alternative 20: 51.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -2.5 \cdot 10^{-25} \lor \neg \left(Vef \leq 3 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= Vef -2.5e-25) (not (<= Vef 3e+96)))
   (/ NaChar (+ (exp (/ (- Vef mu) KbT)) 1.0))
   (/ NaChar (+ (exp (/ (- EAccept mu) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((Vef <= -2.5e-25) || !(Vef <= 3e+96)) {
		tmp = NaChar / (exp(((Vef - mu) / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((vef <= (-2.5d-25)) .or. (.not. (vef <= 3d+96))) then
        tmp = nachar / (exp(((vef - mu) / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp(((eaccept - mu) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((Vef <= -2.5e-25) || !(Vef <= 3e+96)) {
		tmp = NaChar / (Math.exp(((Vef - mu) / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp(((EAccept - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (Vef <= -2.5e-25) or not (Vef <= 3e+96):
		tmp = NaChar / (math.exp(((Vef - mu) / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp(((EAccept - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((Vef <= -2.5e-25) || !(Vef <= 3e+96))
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Vef - mu) / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((Vef <= -2.5e-25) || ~((Vef <= 3e+96)))
		tmp = NaChar / (exp(((Vef - mu) / KbT)) + 1.0);
	else
		tmp = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[Vef, -2.5e-25], N[Not[LessEqual[Vef, 3e+96]], $MachinePrecision]], N[(NaChar / N[(N[Exp[N[(N[(Vef - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(EAccept - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Vef < -2.49999999999999981e-25 or 3e96 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in Ev around 0

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in EAccept around 0

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

   11. Applied rewrites 67.7%

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

   if -2.49999999999999981e-25 < Vef < 3e96

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around 0

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

      1. 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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in Ev around 0

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

   8. Applied rewrites 54.1%

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

   9. Taylor expanded in Vef around 0

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

   11. Applied rewrites 54.8%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -2.5 \cdot 10^{-25} \lor \neg \left(Vef \leq 3 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 45.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -4.8 \cdot 10^{+77} \lor \neg \left(NaChar \leq 3.8 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{-Ec}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -4.8e+77) (not (<= NaChar 3.8e-116)))
   (/ NaChar (+ (exp (/ (- EAccept mu) KbT)) 1.0))
   (/ NdChar (+ (exp (/ (- Ec) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -4.8e+77) || !(NaChar <= 3.8e-116)) {
		tmp = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (exp((-Ec / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((nachar <= (-4.8d+77)) .or. (.not. (nachar <= 3.8d-116))) then
        tmp = nachar / (exp(((eaccept - mu) / kbt)) + 1.0d0)
    else
        tmp = ndchar / (exp((-ec / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -4.8e+77) || !(NaChar <= 3.8e-116)) {
		tmp = NaChar / (Math.exp(((EAccept - mu) / KbT)) + 1.0);
	} else {
		tmp = NdChar / (Math.exp((-Ec / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -4.8e+77) or not (NaChar <= 3.8e-116):
		tmp = NaChar / (math.exp(((EAccept - mu) / KbT)) + 1.0)
	else:
		tmp = NdChar / (math.exp((-Ec / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -4.8e+77) || !(NaChar <= 3.8e-116))
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept - mu) / KbT)) + 1.0));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(-Ec) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -4.8e+77) || ~((NaChar <= 3.8e-116)))
		tmp = NaChar / (exp(((EAccept - mu) / KbT)) + 1.0);
	else
		tmp = NdChar / (exp((-Ec / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -4.8e+77], N[Not[LessEqual[NaChar, 3.8e-116]], $MachinePrecision]], N[(NaChar / N[(N[Exp[N[(N[(EAccept - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -4.7999999999999997e77 or 3.8000000000000001e-116 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in Ev around 0

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

   8. Applied rewrites 72.3%

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

   9. Taylor expanded in Vef around 0

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

   11. Applied rewrites 58.9%

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

   if -4.7999999999999997e77 < NaChar < 3.8000000000000001e-116

   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 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 NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. 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 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in Ec around inf

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

   10. Applied rewrites 45.9%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -4.8 \cdot 10^{+77} \lor \neg \left(NaChar \leq 3.8 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{-Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 22.4% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -1.22 \cdot 10^{+97} \lor \neg \left(NaChar \leq 2.05 \cdot 10^{-147}\right):\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -1.22e+97) (not (<= NaChar 2.05e-147)))
   (* 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 ((NaChar <= -1.22e+97) || !(NaChar <= 2.05e-147)) {
		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 ((nachar <= (-1.22d+97)) .or. (.not. (nachar <= 2.05d-147))) 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 ((NaChar <= -1.22e+97) || !(NaChar <= 2.05e-147)) {
		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 (NaChar <= -1.22e+97) or not (NaChar <= 2.05e-147):
		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 ((NaChar <= -1.22e+97) || !(NaChar <= 2.05e-147))
		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 ((NaChar <= -1.22e+97) || ~((NaChar <= 2.05e-147)))
		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[Or[LessEqual[NaChar, -1.22e+97], N[Not[LessEqual[NaChar, 2.05e-147]], $MachinePrecision]], N[(0.5 * NaChar), $MachinePrecision], N[(0.5 * NdChar), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.21999999999999997e97 or 2.05e-147 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 30.2

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 28.0%

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

   if -1.21999999999999997e97 < NaChar < 2.05e-147

   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. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 22.5

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

   5. Applied rewrites 22.5%

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

   6. Taylor expanded in NdChar around inf

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

   8. Applied rewrites 21.1%

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

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.22 \cdot 10^{+97} \lor \neg \left(NaChar \leq 2.05 \cdot 10^{-147}\right):\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 27.3% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 26.3

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

5. Applied rewrites 26.3%

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

Alternative 24: 18.5% 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 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 26.3

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

5. Applied rewrites 26.3%

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

6. Taylor expanded in NdChar around 0

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

8. Applied rewrites 19.1%

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

Reproduce

herbie shell --seed 2024315 
(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))))))