Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 14.1s

Alternatives: 19

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = -1.1877389875649328e-200
  2. Ec = -3.0531977028876484e+250
  3. Vef = -1.1351512746523687e+304
  4. EDonor = 7.164792657282434e+146
  5. mu = -1.098877102298438e-113
  6. KbT = 8.220913109238178e-301
  7. NaChar = -2.8191171456521906e-202
  8. Ev = 5.690698996452843e-226
  9. EAccept = -1.048073043265885e-89

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 40.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-239} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{\frac{Ev + Vef}{EAccept} + 1}{KbT} \cdot EAccept}\\ \end{array} \end{array} \]

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 -4e-239) (not (<= t_0 2e-230)))
     (* 0.5 (+ NaChar NdChar))
     (/ NaChar (+ 2.0 (* (/ (+ (/ (+ Ev Vef) EAccept) 1.0) KbT) EAccept))))))

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 <= -4e-239) || !(t_0 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (2.0 + (((((Ev + Vef) / EAccept) + 1.0) / KbT) * EAccept));
	}
	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 <= (-4d-239)) .or. (.not. (t_0 <= 2d-230))) then
        tmp = 0.5d0 * (nachar + ndchar)
    else
        tmp = nachar / (2.0d0 + (((((ev + vef) / eaccept) + 1.0d0) / kbt) * eaccept))
    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 <= -4e-239) || !(t_0 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (2.0 + (((((Ev + Vef) / EAccept) + 1.0) / KbT) * EAccept));
	}
	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 <= -4e-239) or not (t_0 <= 2e-230):
		tmp = 0.5 * (NaChar + NdChar)
	else:
		tmp = NaChar / (2.0 + (((((Ev + Vef) / EAccept) + 1.0) / KbT) * EAccept))
	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 <= -4e-239) || !(t_0 <= 2e-230))
		tmp = Float64(0.5 * Float64(NaChar + NdChar));
	else
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(Float64(Float64(Ev + Vef) / EAccept) + 1.0) / KbT) * EAccept)));
	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 <= -4e-239) || ~((t_0 <= 2e-230)))
		tmp = 0.5 * (NaChar + NdChar);
	else
		tmp = NaChar / (2.0 + (((((Ev + Vef) / EAccept) + 1.0) / KbT) * EAccept));
	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, -4e-239], N[Not[LessEqual[t$95$0, 2e-230]], $MachinePrecision]], N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(2.0 + N[(N[(N[(N[(N[(Ev + Vef), $MachinePrecision] / EAccept), $MachinePrecision] + 1.0), $MachinePrecision] / KbT), $MachinePrecision] * EAccept), $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))))) < -4.0000000000000003e-239 or 2.00000000000000009e-230 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 34.7

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

   5. Applied rewrites 34.7%

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

   if -4.0000000000000003e-239 < (+.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))))) < 2.00000000000000009e-230

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 54.9%

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

   9. Taylor expanded in EAccept around inf

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

   11. Applied rewrites 54.4%

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

   12. Taylor expanded in mu around 0

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

   14. Applied rewrites 65.7%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -4 \cdot 10^{-239} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{\frac{Ev + Vef}{EAccept} + 1}{KbT} \cdot EAccept}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{t\_0}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-210} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ (- (+ (+ Ev Vef) EAccept) mu) KbT))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp t_0))))))
   (if (or (<= t_1 -2e-210) (not (<= t_1 2e-230)))
     (* 0.5 (+ NaChar NdChar))
     (/ NaChar (+ 2.0 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 = (((Ev + Vef) + EAccept) - mu) / KbT;
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(t_0)));
	double tmp;
	if ((t_1 <= -2e-210) || !(t_1 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (2.0 + 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 = (((ev + vef) + eaccept) - mu) / kbt
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(t_0)))
    if ((t_1 <= (-2d-210)) .or. (.not. (t_1 <= 2d-230))) then
        tmp = 0.5d0 * (nachar + ndchar)
    else
        tmp = nachar / (2.0d0 + 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 = (((Ev + Vef) + EAccept) - mu) / KbT;
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(t_0)));
	double tmp;
	if ((t_1 <= -2e-210) || !(t_1 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (2.0 + t_0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (((Ev + Vef) + EAccept) - mu) / KbT
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(t_0)))
	tmp = 0
	if (t_1 <= -2e-210) or not (t_1 <= 2e-230):
		tmp = 0.5 * (NaChar + NdChar)
	else:
		tmp = NaChar / (2.0 + t_0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 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 + exp(t_0))))
	tmp = 0.0
	if ((t_1 <= -2e-210) || !(t_1 <= 2e-230))
		tmp = Float64(0.5 * Float64(NaChar + NdChar));
	else
		tmp = Float64(NaChar / Float64(2.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (((Ev + Vef) + EAccept) - mu) / KbT;
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(t_0)));
	tmp = 0.0;
	if ((t_1 <= -2e-210) || ~((t_1 <= 2e-230)))
		tmp = 0.5 * (NaChar + NdChar);
	else
		tmp = NaChar / (2.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $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[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-210], N[Not[LessEqual[t$95$1, 2e-230]], $MachinePrecision]], N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(2.0 + t$95$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))))) < -2.0000000000000001e-210 or 2.00000000000000009e-230 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 35.0

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

   5. Applied rewrites 35.0%

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

   if -2.0000000000000001e-210 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.00000000000000009e-230

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 52.3%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-210} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-210} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{\left(EAccept + Vef\right) - mu}{KbT}}\\ \end{array} \end{array} \]

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))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-210) (not (<= t_0 2e-230)))
     (* 0.5 (+ NaChar NdChar))
     (/ NaChar (+ 2.0 (/ (- (+ EAccept 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-210) || !(t_0 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (2.0 + (((EAccept + 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-210)) .or. (.not. (t_0 <= 2d-230))) then
        tmp = 0.5d0 * (nachar + ndchar)
    else
        tmp = nachar / (2.0d0 + (((eaccept + 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-210) || !(t_0 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (2.0 + (((EAccept + 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-210) or not (t_0 <= 2e-230):
		tmp = 0.5 * (NaChar + NdChar)
	else:
		tmp = NaChar / (2.0 + (((EAccept + 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-210) || !(t_0 <= 2e-230))
		tmp = Float64(0.5 * Float64(NaChar + NdChar));
	else
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(Float64(EAccept + 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-210) || ~((t_0 <= 2e-230)))
		tmp = 0.5 * (NaChar + NdChar);
	else
		tmp = NaChar / (2.0 + (((EAccept + 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-210], N[Not[LessEqual[t$95$0, 2e-230]], $MachinePrecision]], N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(2.0 + N[(N[(N[(EAccept + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $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))))) < -2.0000000000000001e-210 or 2.00000000000000009e-230 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 35.0

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

   5. Applied rewrites 35.0%

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

   if -2.0000000000000001e-210 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.00000000000000009e-230

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 52.3%

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

   9. Taylor expanded in Ev around 0

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

   11. Applied rewrites 45.2%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-210} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{\left(EAccept + Vef\right) - mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 33.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\frac{Vef}{KbT}}\\ \end{array} \end{array} \]

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))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-291) (not (<= t_0 2e-230)))
     (* 0.5 (+ NaChar NdChar))
     (/ NaChar (/ Vef KbT)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-291) || !(t_0 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (Vef / 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-291)) .or. (.not. (t_0 <= 2d-230))) then
        tmp = 0.5d0 * (nachar + ndchar)
    else
        tmp = nachar / (vef / 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-291) || !(t_0 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (Vef / 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-291) or not (t_0 <= 2e-230):
		tmp = 0.5 * (NaChar + NdChar)
	else:
		tmp = NaChar / (Vef / 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-291) || !(t_0 <= 2e-230))
		tmp = Float64(0.5 * Float64(NaChar + NdChar));
	else
		tmp = Float64(NaChar / Float64(Vef / 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-291) || ~((t_0 <= 2e-230)))
		tmp = 0.5 * (NaChar + NdChar);
	else
		tmp = NaChar / (Vef / 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-291], N[Not[LessEqual[t$95$0, 2e-230]], $MachinePrecision]], N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(Vef / KbT), $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))))) < -1.99999999999999992e-291 or 2.00000000000000009e-230 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 34.1

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

   5. Applied rewrites 34.1%

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

   if -1.99999999999999992e-291 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.00000000000000009e-230

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 55.1%

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

   9. Taylor expanded in Vef around inf

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

   11. Applied rewrites 34.3%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-291} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\frac{Vef}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-263} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\frac{EAccept}{KbT}}\\ \end{array} \end{array} \]

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))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-263) (not (<= t_0 2e-230)))
     (* 0.5 (+ NaChar NdChar))
     (/ NaChar (/ 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)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-263) || !(t_0 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (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) :: 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-263)) .or. (.not. (t_0 <= 2d-230))) then
        tmp = 0.5d0 * (nachar + ndchar)
    else
        tmp = nachar / (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)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-263) || !(t_0 <= 2e-230)) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (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)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -2e-263) or not (t_0 <= 2e-230):
		tmp = 0.5 * (NaChar + NdChar)
	else:
		tmp = NaChar / (EAccept / 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-263) || !(t_0 <= 2e-230))
		tmp = Float64(0.5 * Float64(NaChar + NdChar));
	else
		tmp = Float64(NaChar / 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)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -2e-263) || ~((t_0 <= 2e-230)))
		tmp = 0.5 * (NaChar + NdChar);
	else
		tmp = NaChar / (EAccept / 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-263], N[Not[LessEqual[t$95$0, 2e-230]], $MachinePrecision]], N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(EAccept / KbT), $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-263 or 2.00000000000000009e-230 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 34.2

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

   5. Applied rewrites 34.2%

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

   if -2e-263 < (+.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))))) < 2.00000000000000009e-230

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in EAccept around inf

      \[\leadsto \frac{NaChar}{\frac{EAccept}{KbT}} \]
   10. Step-by-step derivation

   11. Applied rewrites 22.1%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-263} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-230}\right):\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\frac{EAccept}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + {\left(e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}\right)}^{-1}}\\ \mathbf{if}\;NdChar \leq -3.1 \cdot 10^{-85}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + \left(1 + Ev \cdot \left(\frac{\frac{\left(EAccept + Vef\right) - mu}{KbT}}{Ev} - \frac{-1}{KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq 5.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + \left(1 + EAccept \cdot \left(\frac{\frac{\left(Ev + Vef\right) - mu}{KbT}}{EAccept} - \frac{-1}{KbT}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (/
          NdChar
          (+ 1.0 (pow (exp (/ (- (- (- Ec Vef) EDonor) mu) KbT)) -1.0)))))
   (if (<= NdChar -3.1e-85)
     (+
      t_0
      (/
       NaChar
       (+
        1.0
        (+
         1.0
         (* Ev (- (/ (/ (- (+ EAccept Vef) mu) KbT) Ev) (/ -1.0 KbT)))))))
     (if (<= NdChar 5.4e+155)
       (/ NaChar (+ (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) 1.0))
       (+
        t_0
        (/
         NaChar
         (+
          1.0
          (+
           1.0
           (*
            EAccept
            (- (/ (/ (- (+ Ev Vef) mu) KbT) EAccept) (/ -1.0 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 + pow(exp(((((Ec - Vef) - EDonor) - mu) / KbT)), -1.0));
	double tmp;
	if (NdChar <= -3.1e-85) {
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (Ev * (((((EAccept + Vef) - mu) / KbT) / Ev) - (-1.0 / KbT))))));
	} else if (NdChar <= 5.4e+155) {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else {
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (EAccept * (((((Ev + Vef) - mu) / KbT) / EAccept) - (-1.0 / 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)) ** (-1.0d0)))
    if (ndchar <= (-3.1d-85)) then
        tmp = t_0 + (nachar / (1.0d0 + (1.0d0 + (ev * (((((eaccept + vef) - mu) / kbt) / ev) - ((-1.0d0) / kbt))))))
    else if (ndchar <= 5.4d+155) then
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) + 1.0d0)
    else
        tmp = t_0 + (nachar / (1.0d0 + (1.0d0 + (eaccept * (((((ev + vef) - mu) / kbt) / eaccept) - ((-1.0d0) / 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.pow(Math.exp(((((Ec - Vef) - EDonor) - mu) / KbT)), -1.0));
	double tmp;
	if (NdChar <= -3.1e-85) {
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (Ev * (((((EAccept + Vef) - mu) / KbT) / Ev) - (-1.0 / KbT))))));
	} else if (NdChar <= 5.4e+155) {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else {
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (EAccept * (((((Ev + Vef) - mu) / KbT) / EAccept) - (-1.0 / KbT))))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.pow(math.exp(((((Ec - Vef) - EDonor) - mu) / KbT)), -1.0))
	tmp = 0
	if NdChar <= -3.1e-85:
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (Ev * (((((EAccept + Vef) - mu) / KbT) / Ev) - (-1.0 / KbT))))))
	elif NdChar <= 5.4e+155:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0)
	else:
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (EAccept * (((((Ev + Vef) - mu) / KbT) / EAccept) - (-1.0 / 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(Ec - Vef) - EDonor) - mu) / KbT)) ^ -1.0)))
	tmp = 0.0
	if (NdChar <= -3.1e-85)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev * Float64(Float64(Float64(Float64(Float64(EAccept + Vef) - mu) / KbT) / Ev) - Float64(-1.0 / KbT)))))));
	elseif (NdChar <= 5.4e+155)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) + 1.0));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(EAccept * Float64(Float64(Float64(Float64(Float64(Ev + Vef) - mu) / KbT) / EAccept) - Float64(-1.0 / 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)) ^ -1.0));
	tmp = 0.0;
	if (NdChar <= -3.1e-85)
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (Ev * (((((EAccept + Vef) - mu) / KbT) / Ev) - (-1.0 / KbT))))));
	elseif (NdChar <= 5.4e+155)
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	else
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (EAccept * (((((Ev + Vef) - mu) / KbT) / EAccept) - (-1.0 / 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[Power[N[Exp[N[(N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -3.1e-85], N[(t$95$0 + N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev * N[(N[(N[(N[(N[(EAccept + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision] / Ev), $MachinePrecision] - N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 5.4e+155], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar / N[(1.0 + N[(1.0 + N[(EAccept * N[(N[(N[(N[(N[(Ev + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision] / EAccept), $MachinePrecision] - N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -3.1000000000000002e-85

   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. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in KbT around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-add N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in Ev around -inf

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

   10. Applied rewrites 71.9%

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

   if -3.1000000000000002e-85 < NdChar < 5.39999999999999987e155

   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 78.4

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

   5. Applied rewrites 78.4%

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

   if 5.39999999999999987e155 < NdChar

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in KbT around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-add N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 80.5

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

   7. Applied rewrites 80.5%

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

   8. Taylor expanded in EAccept around -inf

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

   10. Applied rewrites 83.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -3.1 \cdot 10^{-85}:\\ \;\;\;\;\frac{NdChar}{1 + {\left(e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}\right)}^{-1}} + \frac{NaChar}{1 + \left(1 + Ev \cdot \left(\frac{\frac{\left(EAccept + Vef\right) - mu}{KbT}}{Ev} - \frac{-1}{KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq 5.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + {\left(e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}\right)}^{-1}} + \frac{NaChar}{1 + \left(1 + EAccept \cdot \left(\frac{\frac{\left(Ev + Vef\right) - mu}{KbT}}{EAccept} - \frac{-1}{KbT}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 5.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + {\left(e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}\right)}^{-1}} + \frac{NaChar}{1 + \left(1 + EAccept \cdot \left(\frac{\frac{\left(Ev + Vef\right) - mu}{KbT}}{EAccept} - \frac{-1}{KbT}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -2.8e-67)
   (/ NdChar (+ (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) 1.0))
   (if (<= NdChar 5.4e+155)
     (/ NaChar (+ (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) 1.0))
     (+
      (/ NdChar (+ 1.0 (pow (exp (/ (- (- (- Ec Vef) EDonor) mu) KbT)) -1.0)))
      (/
       NaChar
       (+
        1.0
        (+
         1.0
         (*
          EAccept
          (- (/ (/ (- (+ Ev Vef) mu) KbT) EAccept) (/ -1.0 KbT))))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -2.8e-67) {
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else if (NdChar <= 5.4e+155) {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else {
		tmp = (NdChar / (1.0 + pow(exp(((((Ec - Vef) - EDonor) - mu) / KbT)), -1.0))) + (NaChar / (1.0 + (1.0 + (EAccept * (((((Ev + Vef) - mu) / KbT) / EAccept) - (-1.0 / 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) :: tmp
    if (ndchar <= (-2.8d-67)) then
        tmp = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) + 1.0d0)
    else if (ndchar <= 5.4d+155) then
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) + 1.0d0)
    else
        tmp = (ndchar / (1.0d0 + (exp(((((ec - vef) - edonor) - mu) / kbt)) ** (-1.0d0)))) + (nachar / (1.0d0 + (1.0d0 + (eaccept * (((((ev + vef) - mu) / kbt) / eaccept) - ((-1.0d0) / 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 tmp;
	if (NdChar <= -2.8e-67) {
		tmp = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else if (NdChar <= 5.4e+155) {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	} else {
		tmp = (NdChar / (1.0 + Math.pow(Math.exp(((((Ec - Vef) - EDonor) - mu) / KbT)), -1.0))) + (NaChar / (1.0 + (1.0 + (EAccept * (((((Ev + Vef) - mu) / KbT) / EAccept) - (-1.0 / KbT))))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -2.8e-67:
		tmp = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0)
	elif NdChar <= 5.4e+155:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0)
	else:
		tmp = (NdChar / (1.0 + math.pow(math.exp(((((Ec - Vef) - EDonor) - mu) / KbT)), -1.0))) + (NaChar / (1.0 + (1.0 + (EAccept * (((((Ev + Vef) - mu) / KbT) / EAccept) - (-1.0 / KbT))))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -2.8e-67)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) + 1.0));
	elseif (NdChar <= 5.4e+155)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) + 1.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + (exp(Float64(Float64(Float64(Float64(Ec - Vef) - EDonor) - mu) / KbT)) ^ -1.0))) + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(EAccept * Float64(Float64(Float64(Float64(Float64(Ev + Vef) - mu) / KbT) / EAccept) - Float64(-1.0 / KbT)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -2.8e-67)
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	elseif (NdChar <= 5.4e+155)
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	else
		tmp = (NdChar / (1.0 + (exp(((((Ec - Vef) - EDonor) - mu) / KbT)) ^ -1.0))) + (NaChar / (1.0 + (1.0 + (EAccept * (((((Ev + Vef) - mu) / KbT) / EAccept) - (-1.0 / KbT))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -2.8e-67], N[(NdChar / N[(N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 5.4e+155], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Power[N[Exp[N[(N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(1.0 + N[(EAccept * N[(N[(N[(N[(N[(Ev + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision] / EAccept), $MachinePrecision] - N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -2.8000000000000001e-67

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in EAccept around 0

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

   8. Applied rewrites 73.0%

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

   9. Taylor expanded in NdChar around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-+.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 71.7

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

   11. Applied rewrites 71.7%

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

   if -2.8000000000000001e-67 < NdChar < 5.39999999999999987e155

   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 78.3

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

   5. Applied rewrites 78.3%

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

   if 5.39999999999999987e155 < NdChar

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in KbT around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-add N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 80.5

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

   7. Applied rewrites 80.5%

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

   8. Taylor expanded in EAccept around -inf

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

   10. Applied rewrites 83.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 5.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + {\left(e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}\right)}^{-1}} + \frac{NaChar}{1 + \left(1 + EAccept \cdot \left(\frac{\frac{\left(Ev + Vef\right) - mu}{KbT}}{EAccept} - \frac{-1}{KbT}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Ev + Vef\right) + EAccept\\ \mathbf{if}\;mu \leq -3.4 \cdot 10^{+139} \lor \neg \left(mu \leq 1.85 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{t\_0 - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{t\_0}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ Ev Vef) EAccept)))
   (if (or (<= mu -3.4e+139) (not (<= mu 1.85e+146)))
     (+
      (/ NdChar (+ 1.0 (exp (/ mu KbT))))
      (/ NaChar (+ 1.0 (exp (/ (- t_0 mu) KbT)))))
     (+
      (/ NdChar (+ (exp (/ (- (+ EDonor Vef) Ec) KbT)) 1.0))
      (/ NaChar (+ (exp (/ t_0 KbT)) 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Ev + Vef) + EAccept;
	double tmp;
	if ((mu <= -3.4e+139) || !(mu <= 1.85e+146)) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp(((t_0 - mu) / KbT))));
	} else {
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((t_0 / KbT)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ev + vef) + eaccept
    if ((mu <= (-3.4d+139)) .or. (.not. (mu <= 1.85d+146))) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp(((t_0 - mu) / kbt))))
    else
        tmp = (ndchar / (exp((((edonor + vef) - ec) / kbt)) + 1.0d0)) + (nachar / (exp((t_0 / kbt)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Ev + Vef) + EAccept;
	double tmp;
	if ((mu <= -3.4e+139) || !(mu <= 1.85e+146)) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp(((t_0 - mu) / KbT))));
	} else {
		tmp = (NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp((t_0 / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (Ev + Vef) + EAccept
	tmp = 0
	if (mu <= -3.4e+139) or not (mu <= 1.85e+146):
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp(((t_0 - mu) / KbT))))
	else:
		tmp = (NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp((t_0 / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Ev + Vef) + EAccept)
	tmp = 0.0
	if ((mu <= -3.4e+139) || !(mu <= 1.85e+146))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(t_0 - mu) / KbT)))));
	else
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(t_0 / KbT)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (Ev + Vef) + EAccept;
	tmp = 0.0;
	if ((mu <= -3.4e+139) || ~((mu <= 1.85e+146)))
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp(((t_0 - mu) / KbT))));
	else
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp((t_0 / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision]}, If[Or[LessEqual[mu, -3.4e+139], N[Not[LessEqual[mu, 1.85e+146]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(t$95$0 - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(t$95$0 / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if mu < -3.4000000000000002e139 or 1.85000000000000002e146 < mu

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around inf

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

      1. lower-/.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if -3.4000000000000002e139 < mu < 1.85000000000000002e146

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 97.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -3.4 \cdot 10^{+139} \lor \neg \left(mu \leq 1.85 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;mu \leq -1.4 \cdot 10^{+84} \lor \neg \left(mu \leq 1.5 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Ev + Vef}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= mu -1.4e+84) (not (<= mu 1.5e+146)))
   (+
    (/ NdChar (+ 1.0 (exp (/ mu KbT))))
    (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))
   (+
    (/ NdChar (+ (exp (/ (- (+ EDonor Vef) Ec) KbT)) 1.0))
    (/ NaChar (+ (exp (/ (+ Ev Vef) 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 ((mu <= -1.4e+84) || !(mu <= 1.5e+146)) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	} else {
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((Ev + Vef) / 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 ((mu <= (-1.4d+84)) .or. (.not. (mu <= 1.5d+146))) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    else
        tmp = (ndchar / (exp((((edonor + vef) - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((ev + vef) / 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 ((mu <= -1.4e+84) || !(mu <= 1.5e+146)) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	} else {
		tmp = (NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((Ev + Vef) / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (mu <= -1.4e+84) or not (mu <= 1.5e+146):
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	else:
		tmp = (NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((Ev + Vef) / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((mu <= -1.4e+84) || !(mu <= 1.5e+146))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))));
	else
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Ev + Vef) / 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 ((mu <= -1.4e+84) || ~((mu <= 1.5e+146)))
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	else
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((Ev + Vef) / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[mu, -1.4e+84], N[Not[LessEqual[mu, 1.5e+146]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / 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], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(Ev + Vef), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if mu < -1.39999999999999991e84 or 1.50000000000000001e146 < mu

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around inf

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

      1. lower-/.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if -1.39999999999999991e84 < mu < 1.50000000000000001e146

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in EAccept around 0

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

   8. Applied rewrites 88.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.4 \cdot 10^{+84} \lor \neg \left(mu \leq 1.5 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Ev + Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 12: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EDonor \leq -1.7 \cdot 10^{+34} \lor \neg \left(EDonor \leq 4.6 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Ev + Vef}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= EDonor -1.7e+34) (not (<= EDonor 4.6e+173)))
   (+
    (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
    (/ NaChar (+ (exp (/ (+ (+ Ev Vef) EAccept) KbT)) 1.0)))
   (+
    (/ NdChar (+ (exp (/ (- Vef Ec) KbT)) 1.0))
    (/ NaChar (+ (exp (/ (+ Ev Vef) 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 ((EDonor <= -1.7e+34) || !(EDonor <= 4.6e+173)) {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	} else {
		tmp = (NdChar / (exp(((Vef - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((Ev + Vef) / 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 ((edonor <= (-1.7d+34)) .or. (.not. (edonor <= 4.6d+173))) then
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar / (exp((((ev + vef) + eaccept) / kbt)) + 1.0d0))
    else
        tmp = (ndchar / (exp(((vef - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((ev + vef) / 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 ((EDonor <= -1.7e+34) || !(EDonor <= 4.6e+173)) {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar / (Math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	} else {
		tmp = (NdChar / (Math.exp(((Vef - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((Ev + Vef) / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (EDonor <= -1.7e+34) or not (EDonor <= 4.6e+173):
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar / (math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0))
	else:
		tmp = (NdChar / (math.exp(((Vef - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((Ev + Vef) / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((EDonor <= -1.7e+34) || !(EDonor <= 4.6e+173))
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + Vef) + EAccept) / KbT)) + 1.0)));
	else
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Vef - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Ev + Vef) / 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 ((EDonor <= -1.7e+34) || ~((EDonor <= 4.6e+173)))
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	else
		tmp = (NdChar / (exp(((Vef - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((Ev + Vef) / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[EDonor, -1.7e+34], N[Not[LessEqual[EDonor, 4.6e+173]], $MachinePrecision]], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[Exp[N[(N[(Vef - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(Ev + Vef), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if EDonor < -1.7e34 or 4.5999999999999999e173 < EDonor

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 87.4%

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

   if -1.7e34 < EDonor < 4.5999999999999999e173

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in EAccept around 0

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

   8. Applied rewrites 78.2%

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

   9. Taylor expanded in EDonor around 0

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

   11. Applied rewrites 77.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -1.7 \cdot 10^{+34} \lor \neg \left(EDonor \leq 4.6 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Ev + Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq 6.5 \cdot 10^{+198}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Ev + Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept 6.5e+198)
   (+
    (/ NdChar (+ (exp (/ (- (+ EDonor Vef) Ec) KbT)) 1.0))
    (/ NaChar (+ (exp (/ (+ Ev Vef) KbT)) 1.0)))
   (+
    (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
    (/ NaChar (+ (exp (/ (+ (+ Ev Vef) EAccept) KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 6.5e+198) {
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((Ev + Vef) / KbT)) + 1.0));
	} else {
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (eaccept <= 6.5d+198) then
        tmp = (ndchar / (exp((((edonor + vef) - ec) / kbt)) + 1.0d0)) + (nachar / (exp(((ev + vef) / kbt)) + 1.0d0))
    else
        tmp = (ndchar / (exp((edonor / kbt)) + 1.0d0)) + (nachar / (exp((((ev + vef) + eaccept) / kbt)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 6.5e+198) {
		tmp = (NdChar / (Math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (Math.exp(((Ev + Vef) / KbT)) + 1.0));
	} else {
		tmp = (NdChar / (Math.exp((EDonor / KbT)) + 1.0)) + (NaChar / (Math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EAccept <= 6.5e+198:
		tmp = (NdChar / (math.exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (math.exp(((Ev + Vef) / KbT)) + 1.0))
	else:
		tmp = (NdChar / (math.exp((EDonor / KbT)) + 1.0)) + (NaChar / (math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EAccept <= 6.5e+198)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Ev + Vef) / KbT)) + 1.0)));
	else
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + Vef) + EAccept) / KbT)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EAccept <= 6.5e+198)
		tmp = (NdChar / (exp((((EDonor + Vef) - Ec) / KbT)) + 1.0)) + (NaChar / (exp(((Ev + Vef) / KbT)) + 1.0));
	else
		tmp = (NdChar / (exp((EDonor / KbT)) + 1.0)) + (NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EAccept, 6.5e+198], N[(N[(NdChar / N[(N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(Ev + Vef), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if EAccept < 6.5000000000000003e198

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in EAccept around 0

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

   8. Applied rewrites 81.1%

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

   if 6.5000000000000003e198 < 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 mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in EDonor around inf

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

   8. Applied rewrites 83.7%

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

Alternative 14: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.8 \cdot 10^{-67} \lor \neg \left(NdChar \leq 2 \cdot 10^{+213}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -2.8e-67) (not (<= NdChar 2e+213)))
   (/ NdChar (+ (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) 1.0))
   (/ NaChar (+ (exp (/ (- (+ (+ Ev Vef) 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 ((NdChar <= -2.8e-67) || !(NdChar <= 2e+213)) {
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp(((((Ev + Vef) + 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 ((ndchar <= (-2.8d-67)) .or. (.not. (ndchar <= 2d+213))) then
        tmp = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp(((((ev + vef) + 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 ((NdChar <= -2.8e-67) || !(NdChar <= 2e+213)) {
		tmp = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -2.8e-67) or not (NdChar <= 2e+213):
		tmp = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -2.8e-67) || !(NdChar <= 2e+213))
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + 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 ((NdChar <= -2.8e-67) || ~((NdChar <= 2e+213)))
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	else
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -2.8e-67], N[Not[LessEqual[NdChar, 2e+213]], $MachinePrecision]], N[(NdChar / N[(N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.8000000000000001e-67 or 1.99999999999999997e213 < NdChar

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in EAccept around 0

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

   8. Applied rewrites 76.6%

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

   9. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   10. 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 76.4

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

   11. Applied rewrites 76.4%

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

   if -2.8000000000000001e-67 < NdChar < 1.99999999999999997e213

   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 77.6

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

   5. Applied rewrites 77.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.8 \cdot 10^{-67} \lor \neg \left(NdChar \leq 2 \cdot 10^{+213}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -9 \cdot 10^{+196}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -9e+196)
   (* 0.5 (+ NaChar NdChar))
   (/ NaChar (+ (exp (/ (- (+ (+ Ev Vef) 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 (KbT <= -9e+196) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (exp(((((Ev + Vef) + 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 (kbt <= (-9d+196)) then
        tmp = 0.5d0 * (nachar + ndchar)
    else
        tmp = nachar / (exp(((((ev + vef) + 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 (KbT <= -9e+196) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -9e+196:
		tmp = 0.5 * (NaChar + NdChar)
	else:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -9e+196)
		tmp = Float64(0.5 * Float64(NaChar + NdChar));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + 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 (KbT <= -9e+196)
		tmp = 0.5 * (NaChar + NdChar);
	else
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -9e+196], N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -8.99999999999999956e196

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -8.99999999999999956e196 < KbT

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 62.9

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

   5. Applied rewrites 62.9%

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

Alternative 16: 56.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -9 \cdot 10^{+196}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + Vef\right) + EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -9e+196)
   (* 0.5 (+ NaChar NdChar))
   (/ NaChar (+ (exp (/ (+ (+ Ev Vef) EAccept) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -9e+196) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-9d+196)) then
        tmp = 0.5d0 * (nachar + ndchar)
    else
        tmp = nachar / (exp((((ev + vef) + eaccept) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -9e+196) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (Math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -9e+196:
		tmp = 0.5 * (NaChar + NdChar)
	else:
		tmp = NaChar / (math.exp((((Ev + Vef) + EAccept) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -9e+196)
		tmp = Float64(0.5 * Float64(NaChar + NdChar));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + Vef) + EAccept) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -9e+196)
		tmp = 0.5 * (NaChar + NdChar);
	else
		tmp = NaChar / (exp((((Ev + Vef) + EAccept) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -9e+196], N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -8.99999999999999956e196

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -8.99999999999999956e196 < KbT

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in mu around 0

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

   8. Applied rewrites 59.0%

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

Alternative 17: 21.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.9 \cdot 10^{-23} \lor \neg \left(NdChar \leq 3.6 \cdot 10^{+214}\right):\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -2.9e-23) (not (<= NdChar 3.6e+214)))
   (* 0.5 NdChar)
   (* 0.5 NaChar)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.9e-23) || !(NdChar <= 3.6e+214)) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = 0.5 * 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) :: tmp
    if ((ndchar <= (-2.9d-23)) .or. (.not. (ndchar <= 3.6d+214))) then
        tmp = 0.5d0 * ndchar
    else
        tmp = 0.5d0 * 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 tmp;
	if ((NdChar <= -2.9e-23) || !(NdChar <= 3.6e+214)) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = 0.5 * NaChar;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -2.9e-23) or not (NdChar <= 3.6e+214):
		tmp = 0.5 * NdChar
	else:
		tmp = 0.5 * NaChar
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -2.9e-23) || !(NdChar <= 3.6e+214))
		tmp = Float64(0.5 * NdChar);
	else
		tmp = Float64(0.5 * NaChar);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -2.9e-23) || ~((NdChar <= 3.6e+214)))
		tmp = 0.5 * NdChar;
	else
		tmp = 0.5 * NaChar;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -2.9e-23], N[Not[LessEqual[NdChar, 3.6e+214]], $MachinePrecision]], N[(0.5 * NdChar), $MachinePrecision], N[(0.5 * NaChar), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.9000000000000002e-23 or 3.6000000000000001e214 < NdChar

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 24.9

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

   5. Applied rewrites 24.9%

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

   6. Taylor expanded in NdChar around inf

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

   8. Applied rewrites 22.9%

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

   if -2.9000000000000002e-23 < NdChar < 3.6000000000000001e214

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 30.0

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

   5. Applied rewrites 30.0%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 28.5%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.9 \cdot 10^{-23} \lor \neg \left(NdChar \leq 3.6 \cdot 10^{+214}\right):\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NaChar\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.7% accurate, 30.7× speedup

Math

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

FPCore

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

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 + NdChar);
}

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 + ndchar)
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 + NdChar);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 28.2

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

5. Applied rewrites 28.2%

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

Alternative 19: 17.6% 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. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 28.2

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

5. Applied rewrites 28.2%

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

6. Taylor expanded in NdChar around 0

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

8. Applied rewrites 21.6%

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

Reproduce

herbie shell --seed 2024337 
(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))))))