Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 19.3s

Alternatives: 18

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = -6.252999834620949e-72
  2. Ec = 2.245582765480097e+77
  3. Vef = -8.59866984453016e+218
  4. EDonor = 4.883072390376385e-100
  5. mu = -4.5120225310076555e+252
  6. KbT = 1.8280260472875894e-62
  7. NaChar = -6.266311138737945e-72
  8. Ev = -5.4285449856614706e+231
  9. EAccept = -1.7309881929692216e+286

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 52.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + NdChar \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ t_2 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{mu}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NaChar (+ (exp (/ Vef KbT)) 1.0)) (* NdChar 0.5)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0))))
        (t_2 (+ EDonor (- (+ Vef mu) Ec))))
   (if (<= t_1 -4e+30)
     t_0
     (if (<= t_1 -5e-288)
       (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
       (if (<= t_1 0.0)
         (/
          NdChar
          (-
           2.0
           (/
            (fma -0.5 (/ (* t_2 t_2) KbT) (- (- Ec (+ Vef mu)) EDonor))
            KbT)))
         (if (<= t_1 2e-70)
           (/ NdChar (+ (exp (/ mu KbT)) 1.0))
           (if (<= t_1 2e+65)
             (/ NaChar (+ (exp (/ mu (- KbT))) 1.0))
             t_0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (exp((Vef / KbT)) + 1.0)) + (NdChar * 0.5);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double t_2 = EDonor + ((Vef + mu) - Ec);
	double tmp;
	if (t_1 <= -4e+30) {
		tmp = t_0;
	} else if (t_1 <= -5e-288) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_2 * t_2) / KbT), ((Ec - (Vef + mu)) - EDonor)) / KbT));
	} else if (t_1 <= 2e-70) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else if (t_1 <= 2e+65) {
		tmp = NaChar / (exp((mu / -KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)) + Float64(NdChar * 0.5))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	t_2 = Float64(EDonor + Float64(Float64(Vef + mu) - Ec))
	tmp = 0.0
	if (t_1 <= -4e+30)
		tmp = t_0;
	elseif (t_1 <= -5e-288)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Float64(Float64(Ec - Float64(Vef + mu)) - EDonor)) / KbT)));
	elseif (t_1 <= 2e-70)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	elseif (t_1 <= 2e+65)
		tmp = Float64(NaChar / Float64(exp(Float64(mu / Float64(-KbT))) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(EDonor + N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+30], t$95$0, If[LessEqual[t$95$1, -5e-288], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-70], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+65], N[(NaChar / N[(N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.0000000000000001e30 or 2e65 < (+.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 Vef around inf

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

      1. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{NdChar \cdot \frac{1}{2}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} \]

      2. lower-*.f64 57.1

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

   8. Applied rewrites 57.1%

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

   if -4.0000000000000001e30 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000011e-288

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 42.9%

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

   if -5.00000000000000011e-288 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 89.3%

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

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.99999999999999999e-70

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in mu around inf

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

   8. Applied rewrites 52.1%

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

   if 1.99999999999999999e-70 < (+.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))))) < 2e65

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in mu around inf

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

   8. Applied rewrites 61.9%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -4 \cdot 10^{+30}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{mu}{-KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_1 := \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-288}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (- (+ Vef mu) Ec)))
        (t_1 (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_2 -5e-288)
     t_1
     (if (<= t_2 0.0)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- Ec (+ Vef mu)) EDonor)) KbT)))
       (if (<= t_2 2e-70)
         (/ NdChar (+ (exp (/ mu KbT)) 1.0))
         (if (<= t_2 5e+147) t_1 (* 0.5 (+ NdChar NaChar))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + ((Vef + mu) - Ec);
	double t_1 = NaChar / (exp((EAccept / KbT)) + 1.0);
	double t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -5e-288) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((Ec - (Vef + mu)) - EDonor)) / KbT));
	} else if (t_2 <= 2e-70) {
		tmp = NdChar / (exp((mu / KbT)) + 1.0);
	} else if (t_2 <= 5e+147) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Float64(Vef + mu) - Ec))
	t_1 = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -5e-288)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Ec - Float64(Vef + mu)) - EDonor)) / KbT)));
	elseif (t_2 <= 2e-70)
		tmp = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0));
	elseif (t_2 <= 5e+147)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-288], t$95$1, If[LessEqual[t$95$2, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-70], N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+147], t$95$1, N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000011e-288 or 1.99999999999999999e-70 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5.0000000000000002e147

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 41.5%

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

   if -5.00000000000000011e-288 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 89.3%

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

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.99999999999999999e-70

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in mu around inf

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

   8. Applied rewrites 52.1%

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

   if 5.0000000000000002e147 < (+.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 50.5

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

   5. Applied rewrites 50.5%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1}\\ t_1 := t\_0 + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ t_2 := t\_0 + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-288}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0)))
        (t_1 (+ t_0 (/ NaChar (+ (exp (/ Vef KbT)) 1.0))))
        (t_2
         (+
          t_0
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_2 -5e-288)
     t_1
     (if (<= t_2 2e-70)
       (/ NdChar (+ (exp (/ (+ (+ Vef EDonor) (- mu Ec)) KbT)) 1.0))
       (if (<= t_2 2e+65)
         (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))
         t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	double t_1 = t_0 + (NaChar / (exp((Vef / KbT)) + 1.0));
	double t_2 = t_0 + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -5e-288) {
		tmp = t_1;
	} else if (t_2 <= 2e-70) {
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else if (t_2 <= 2e+65) {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)
    t_1 = t_0 + (nachar / (exp((vef / kbt)) + 1.0d0))
    t_2 = t_0 + (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0))
    if (t_2 <= (-5d-288)) then
        tmp = t_1
    else if (t_2 <= 2d-70) then
        tmp = ndchar / (exp((((vef + edonor) + (mu - ec)) / kbt)) + 1.0d0)
    else if (t_2 <= 2d+65) then
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	double t_1 = t_0 + (NaChar / (Math.exp((Vef / KbT)) + 1.0));
	double t_2 = t_0 + (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -5e-288) {
		tmp = t_1;
	} else if (t_2 <= 2e-70) {
		tmp = NdChar / (Math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else if (t_2 <= 2e+65) {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)
	t_1 = t_0 + (NaChar / (math.exp((Vef / KbT)) + 1.0))
	t_2 = t_0 + (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_2 <= -5e-288:
		tmp = t_1
	elif t_2 <= 2e-70:
		tmp = NdChar / (math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0)
	elif t_2 <= 2e+65:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0))
	t_1 = Float64(t_0 + Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)))
	t_2 = Float64(t_0 + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -5e-288)
		tmp = t_1;
	elseif (t_2 <= 2e-70)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Vef + EDonor) + Float64(mu - Ec)) / KbT)) + 1.0));
	elseif (t_2 <= 2e+65)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0);
	t_1 = t_0 + (NaChar / (exp((Vef / KbT)) + 1.0));
	t_2 = t_0 + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_2 <= -5e-288)
		tmp = t_1;
	elseif (t_2 <= 2e-70)
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	elseif (t_2 <= 2e+65)
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-288], t$95$1, If[LessEqual[t$95$2, 2e-70], N[(NdChar / N[(N[Exp[N[(N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+65], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000011e-288 or 2e65 < (+.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 Vef around inf

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

      1. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if -5.00000000000000011e-288 < (+.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))))) < 1.99999999999999999e-70

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if 1.99999999999999999e-70 < (+.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))))) < 2e65

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 82.3

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

   5. Applied rewrites 82.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_1 := \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-288}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-235}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (- (+ Vef mu) Ec)))
        (t_1 (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_2 -5e-288)
     t_1
     (if (<= t_2 2e-235)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- Ec (+ Vef mu)) EDonor)) KbT)))
       (if (<= t_2 5e+147) t_1 (* 0.5 (+ NdChar NaChar)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + ((Vef + mu) - Ec);
	double t_1 = NaChar / (exp((EAccept / KbT)) + 1.0);
	double t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -5e-288) {
		tmp = t_1;
	} else if (t_2 <= 2e-235) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((Ec - (Vef + mu)) - EDonor)) / KbT));
	} else if (t_2 <= 5e+147) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Float64(Vef + mu) - Ec))
	t_1 = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -5e-288)
		tmp = t_1;
	elseif (t_2 <= 2e-235)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Ec - Float64(Vef + mu)) - EDonor)) / KbT)));
	elseif (t_2 <= 5e+147)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-288], t$95$1, If[LessEqual[t$95$2, 2e-235], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+147], t$95$1, N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000011e-288 or 1.9999999999999999e-235 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5.0000000000000002e147

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 38.8%

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

   if -5.00000000000000011e-288 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.9999999999999999e-235

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 79.4%

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

   if 5.0000000000000002e147 < (+.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 50.5

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

   5. Applied rewrites 50.5%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-235}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-294}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{-0.5 \cdot \left(Vef \cdot Vef\right)}{KbT \cdot KbT}}\\ \mathbf{elif}\;t\_1 \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_1 -1e-101)
     t_0
     (if (<= t_1 2e-294)
       (/ NaChar (- 2.0 (/ (* -0.5 (* Vef Vef)) (* KbT KbT))))
       (if (<= t_1 1e-75) (* NdChar 0.5) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-101) {
		tmp = t_0;
	} else if (t_1 <= 2e-294) {
		tmp = NaChar / (2.0 - ((-0.5 * (Vef * Vef)) / (KbT * KbT)));
	} else if (t_1 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-1d-101)) then
        tmp = t_0
    else if (t_1 <= 2d-294) then
        tmp = nachar / (2.0d0 - (((-0.5d0) * (vef * vef)) / (kbt * kbt)))
    else if (t_1 <= 1d-75) then
        tmp = ndchar * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-101) {
		tmp = t_0;
	} else if (t_1 <= 2e-294) {
		tmp = NaChar / (2.0 - ((-0.5 * (Vef * Vef)) / (KbT * KbT)));
	} else if (t_1 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -1e-101:
		tmp = t_0
	elif t_1 <= 2e-294:
		tmp = NaChar / (2.0 - ((-0.5 * (Vef * Vef)) / (KbT * KbT)))
	elif t_1 <= 1e-75:
		tmp = NdChar * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-101)
		tmp = t_0;
	elseif (t_1 <= 2e-294)
		tmp = Float64(NaChar / Float64(2.0 - Float64(Float64(-0.5 * Float64(Vef * Vef)) / Float64(KbT * KbT))));
	elseif (t_1 <= 1e-75)
		tmp = Float64(NdChar * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -1e-101)
		tmp = t_0;
	elseif (t_1 <= 2e-294)
		tmp = NaChar / (2.0 - ((-0.5 * (Vef * Vef)) / (KbT * KbT)));
	elseif (t_1 <= 1e-75)
		tmp = NdChar * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-101], t$95$0, If[LessEqual[t$95$1, 2e-294], N[(NaChar / N[(2.0 - N[(N[(-0.5 * N[(Vef * Vef), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-75], N[(NdChar * 0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.00000000000000005e-101 or 9.9999999999999996e-76 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.1

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

   5. Applied rewrites 37.1%

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

   if -1.00000000000000005e-101 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.00000000000000003e-294

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 62.0%

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

   9. Taylor expanded in Vef around inf

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

   11. Applied rewrites 40.2%

       \[\leadsto \frac{NaChar}{2 - \frac{-0.5 \cdot \left(Vef \cdot Vef\right)}{KbT \cdot \color{blue}{KbT}}} \]

   if 2.00000000000000003e-294 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999996e-76

   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 11.5

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

   5. Applied rewrites 11.5%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 29.2%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-294}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{-0.5 \cdot \left(Vef \cdot Vef\right)}{KbT \cdot KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-303}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{-0.5 \cdot \left(Ev \cdot Ev\right)}{KbT \cdot KbT}}\\ \mathbf{elif}\;t\_1 \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-85)
     t_0
     (if (<= t_1 4e-303)
       (/ NaChar (- 2.0 (/ (* -0.5 (* Ev Ev)) (* KbT KbT))))
       (if (<= t_1 1e-75) (* NdChar 0.5) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-85) {
		tmp = t_0;
	} else if (t_1 <= 4e-303) {
		tmp = NaChar / (2.0 - ((-0.5 * (Ev * Ev)) / (KbT * KbT)));
	} else if (t_1 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-2d-85)) then
        tmp = t_0
    else if (t_1 <= 4d-303) then
        tmp = nachar / (2.0d0 - (((-0.5d0) * (ev * ev)) / (kbt * kbt)))
    else if (t_1 <= 1d-75) then
        tmp = ndchar * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-85) {
		tmp = t_0;
	} else if (t_1 <= 4e-303) {
		tmp = NaChar / (2.0 - ((-0.5 * (Ev * Ev)) / (KbT * KbT)));
	} else if (t_1 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -2e-85:
		tmp = t_0
	elif t_1 <= 4e-303:
		tmp = NaChar / (2.0 - ((-0.5 * (Ev * Ev)) / (KbT * KbT)))
	elif t_1 <= 1e-75:
		tmp = NdChar * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-85)
		tmp = t_0;
	elseif (t_1 <= 4e-303)
		tmp = Float64(NaChar / Float64(2.0 - Float64(Float64(-0.5 * Float64(Ev * Ev)) / Float64(KbT * KbT))));
	elseif (t_1 <= 1e-75)
		tmp = Float64(NdChar * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-85)
		tmp = t_0;
	elseif (t_1 <= 4e-303)
		tmp = NaChar / (2.0 - ((-0.5 * (Ev * Ev)) / (KbT * KbT)));
	elseif (t_1 <= 1e-75)
		tmp = NdChar * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-85], t$95$0, If[LessEqual[t$95$1, 4e-303], N[(NaChar / N[(2.0 - N[(N[(-0.5 * N[(Ev * Ev), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-75], N[(NdChar * 0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-85 or 9.9999999999999996e-76 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.3

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

   5. Applied rewrites 37.3%

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

   if -2e-85 < (+.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))))) < 3.99999999999999972e-303

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in Ev around inf

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

   11. Applied rewrites 38.2%

       \[\leadsto \frac{NaChar}{2 - \frac{-0.5 \cdot \left(Ev \cdot Ev\right)}{KbT \cdot \color{blue}{KbT}}} \]

   if 3.99999999999999972e-303 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999996e-76

   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 10.8

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 27.7%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{-303}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{-0.5 \cdot \left(Ev \cdot Ev\right)}{KbT \cdot KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EAccept + \left(Vef + Ev\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{t\_0 - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-303}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{mu - t\_0}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EAccept (+ Vef Ev)))
        (t_1 (* 0.5 (+ NdChar NaChar)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- t_0 mu) KbT)) 1.0)))))
   (if (<= t_2 -2e-85)
     t_1
     (if (<= t_2 4e-303)
       (/ NaChar (- 2.0 (/ (- mu t_0) KbT)))
       (if (<= t_2 1e-75) (* NdChar 0.5) t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EAccept + (Vef + Ev);
	double t_1 = 0.5 * (NdChar + NaChar);
	double t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((t_0 - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -2e-85) {
		tmp = t_1;
	} else if (t_2 <= 4e-303) {
		tmp = NaChar / (2.0 - ((mu - t_0) / KbT));
	} else if (t_2 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = eaccept + (vef + ev)
    t_1 = 0.5d0 * (ndchar + nachar)
    t_2 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp(((t_0 - mu) / kbt)) + 1.0d0))
    if (t_2 <= (-2d-85)) then
        tmp = t_1
    else if (t_2 <= 4d-303) then
        tmp = nachar / (2.0d0 - ((mu - t_0) / kbt))
    else if (t_2 <= 1d-75) then
        tmp = ndchar * 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EAccept + (Vef + Ev);
	double t_1 = 0.5 * (NdChar + NaChar);
	double t_2 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp(((t_0 - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -2e-85) {
		tmp = t_1;
	} else if (t_2 <= 4e-303) {
		tmp = NaChar / (2.0 - ((mu - t_0) / KbT));
	} else if (t_2 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = EAccept + (Vef + Ev)
	t_1 = 0.5 * (NdChar + NaChar)
	t_2 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp(((t_0 - mu) / KbT)) + 1.0))
	tmp = 0
	if t_2 <= -2e-85:
		tmp = t_1
	elif t_2 <= 4e-303:
		tmp = NaChar / (2.0 - ((mu - t_0) / KbT))
	elif t_2 <= 1e-75:
		tmp = NdChar * 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EAccept + Float64(Vef + Ev))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(t_0 - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -2e-85)
		tmp = t_1;
	elseif (t_2 <= 4e-303)
		tmp = Float64(NaChar / Float64(2.0 - Float64(Float64(mu - t_0) / KbT)));
	elseif (t_2 <= 1e-75)
		tmp = Float64(NdChar * 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = EAccept + (Vef + Ev);
	t_1 = 0.5 * (NdChar + NaChar);
	t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp(((t_0 - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_2 <= -2e-85)
		tmp = t_1;
	elseif (t_2 <= 4e-303)
		tmp = NaChar / (2.0 - ((mu - t_0) / KbT));
	elseif (t_2 <= 1e-75)
		tmp = NdChar * 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(t$95$0 - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-85], t$95$1, If[LessEqual[t$95$2, 4e-303], N[(NaChar / N[(2.0 - N[(N[(mu - t$95$0), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-75], N[(NdChar * 0.5), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-85 or 9.9999999999999996e-76 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.3

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

   5. Applied rewrites 37.3%

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

   if -2e-85 < (+.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))))) < 3.99999999999999972e-303

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in KbT around inf

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

   11. Applied rewrites 39.2%

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

   if 3.99999999999999972e-303 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999996e-76

   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 10.8

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 27.7%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{-303}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{mu - \left(EAccept + \left(Vef + Ev\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-253}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(-EAccept\right) \cdot \frac{-0.25 \cdot \left(NdChar \cdot Ec\right)}{KbT \cdot EAccept}\\ \mathbf{elif}\;t\_1 \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-253)
     t_0
     (if (<= t_1 0.0)
       (* (- EAccept) (/ (* -0.25 (* NdChar Ec)) (* KbT EAccept)))
       (if (<= t_1 1e-75) (* NdChar 0.5) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-253) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = -EAccept * ((-0.25 * (NdChar * Ec)) / (KbT * EAccept));
	} else if (t_1 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-2d-253)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = -eaccept * (((-0.25d0) * (ndchar * ec)) / (kbt * eaccept))
    else if (t_1 <= 1d-75) then
        tmp = ndchar * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-253) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = -EAccept * ((-0.25 * (NdChar * Ec)) / (KbT * EAccept));
	} else if (t_1 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -2e-253:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = -EAccept * ((-0.25 * (NdChar * Ec)) / (KbT * EAccept))
	elif t_1 <= 1e-75:
		tmp = NdChar * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-253)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-EAccept) * Float64(Float64(-0.25 * Float64(NdChar * Ec)) / Float64(KbT * EAccept)));
	elseif (t_1 <= 1e-75)
		tmp = Float64(NdChar * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-253)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = -EAccept * ((-0.25 * (NdChar * Ec)) / (KbT * EAccept));
	elseif (t_1 <= 1e-75)
		tmp = NdChar * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-253], t$95$0, If[LessEqual[t$95$1, 0.0], N[((-EAccept) * N[(N[(-0.25 * N[(NdChar * Ec), $MachinePrecision]), $MachinePrecision] / N[(KbT * EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-75], N[(NdChar * 0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.0000000000000001e-253 or 9.9999999999999996e-76 < (+.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 35.4

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

   5. Applied rewrites 35.4%

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

   if -2.0000000000000001e-253 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in EAccept around -inf

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

   8. Applied rewrites 1.3%

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

   9. Taylor expanded in Ec around inf

      \[\leadsto \left(\mathsf{neg}\left(EAccept\right)\right) \cdot \left(\frac{-1}{4} \cdot \frac{Ec \cdot NdChar}{\color{blue}{EAccept \cdot KbT}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 21.9%

       \[\leadsto \left(-EAccept\right) \cdot \frac{-0.25 \cdot \left(NdChar \cdot Ec\right)}{EAccept \cdot \color{blue}{KbT}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999996e-76

   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 10.2

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 26.5%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\left(-EAccept\right) \cdot \frac{-0.25 \cdot \left(NdChar \cdot Ec\right)}{KbT \cdot EAccept}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-218}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;-0.25 \cdot \frac{NaChar \cdot Ev}{KbT}\\ \mathbf{elif}\;t\_1 \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_1 -5e-218)
     t_0
     (if (<= t_1 0.0)
       (* -0.25 (/ (* NaChar Ev) KbT))
       (if (<= t_1 1e-75) (* NdChar 0.5) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-218) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = -0.25 * ((NaChar * Ev) / KbT);
	} else if (t_1 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + (vef + ev)) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-5d-218)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (-0.25d0) * ((nachar * ev) / kbt)
    else if (t_1 <= 1d-75) then
        tmp = ndchar * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -5e-218) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = -0.25 * ((NaChar * Ev) / KbT);
	} else if (t_1 <= 1e-75) {
		tmp = NdChar * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -5e-218:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = -0.25 * ((NaChar * Ev) / KbT)
	elif t_1 <= 1e-75:
		tmp = NdChar * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -5e-218)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(-0.25 * Float64(Float64(NaChar * Ev) / KbT));
	elseif (t_1 <= 1e-75)
		tmp = Float64(NdChar * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	t_1 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -5e-218)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = -0.25 * ((NaChar * Ev) / KbT);
	elseif (t_1 <= 1e-75)
		tmp = NdChar * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-218], t$95$0, If[LessEqual[t$95$1, 0.0], N[(-0.25 * N[(N[(NaChar * Ev), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-75], N[(NdChar * 0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000041e-218 or 9.9999999999999996e-76 < (+.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 35.7

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

   5. Applied rewrites 35.7%

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

   if -5.00000000000000041e-218 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in Ev around inf

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{Ev \cdot NaChar}{KbT}} \]
   7. Step-by-step derivation

   8. Applied rewrites 10.7%

      \[\leadsto -0.25 \cdot \color{blue}{\frac{NaChar \cdot Ev}{KbT}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999996e-76

   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 10.2

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 26.5%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-218}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;-0.25 \cdot \frac{NaChar \cdot Ev}{KbT}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 10^{-75}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-218}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-233}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (- (+ Vef mu) Ec)))
        (t_1 (* 0.5 (+ NdChar NaChar)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_2 -5e-218)
     t_1
     (if (<= t_2 2e-233)
       (/
        NdChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- Ec (+ Vef mu)) EDonor)) KbT)))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + ((Vef + mu) - Ec);
	double t_1 = 0.5 * (NdChar + NaChar);
	double t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -5e-218) {
		tmp = t_1;
	} else if (t_2 <= 2e-233) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((Ec - (Vef + mu)) - EDonor)) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Float64(Vef + mu) - Ec))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -5e-218)
		tmp = t_1;
	elseif (t_2 <= 2e-233)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(Ec - Float64(Vef + mu)) - EDonor)) / KbT)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-218], t$95$1, If[LessEqual[t$95$2, 2e-233], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000041e-218 or 1.99999999999999992e-233 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   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 33.8

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

   5. Applied rewrites 33.8%

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

   if -5.00000000000000041e-218 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.99999999999999992e-233

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 70.0%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-218}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-233}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Vef - mu\right) + \left(Ev + EAccept\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-233}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(mu - Vef\right) - \left(Ev + EAccept\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (- Vef mu) (+ Ev EAccept)))
        (t_1 (* 0.5 (+ NdChar NaChar)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_2 -2e-85)
     t_1
     (if (<= t_2 2e-233)
       (/
        NaChar
        (-
         2.0
         (/ (fma -0.5 (/ (* t_0 t_0) KbT) (- (- mu Vef) (+ Ev EAccept))) KbT)))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef - mu) + (Ev + EAccept);
	double t_1 = 0.5 * (NdChar + NaChar);
	double t_2 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -2e-85) {
		tmp = t_1;
	} else if (t_2 <= 2e-233) {
		tmp = NaChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((mu - Vef) - (Ev + EAccept))) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Vef - mu) + Float64(Ev + EAccept))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -2e-85)
		tmp = t_1;
	elseif (t_2 <= 2e-233)
		tmp = Float64(NaChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(mu - Vef) - Float64(Ev + EAccept))) / KbT)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Vef - mu), $MachinePrecision] + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-85], t$95$1, If[LessEqual[t$95$2, 2e-233], N[(NaChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(mu - Vef), $MachinePrecision] - N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-85 or 1.99999999999999992e-233 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   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 35.1

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

   5. Applied rewrites 35.1%

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

   if -2e-85 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.99999999999999992e-233

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in KbT around -inf

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

   8. Applied rewrites 56.8%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-85}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-233}:\\ \;\;\;\;\frac{NaChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(\left(Vef - mu\right) + \left(Ev + EAccept\right)\right) \cdot \left(\left(Vef - mu\right) + \left(Ev + EAccept\right)\right)}{KbT}, \left(mu - Vef\right) - \left(Ev + EAccept\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -3.8 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 0.005:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))))
   (if (<= NaChar -3.8e+121)
     t_0
     (if (<= NaChar 0.005)
       (/ NdChar (+ (exp (/ (+ (+ Vef EDonor) (- mu Ec)) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -3.8e+121) {
		tmp = t_0;
	} else if (NaChar <= 0.005) {
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    if (nachar <= (-3.8d+121)) then
        tmp = t_0
    else if (nachar <= 0.005d0) then
        tmp = ndchar / (exp((((vef + edonor) + (mu - ec)) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -3.8e+121) {
		tmp = t_0;
	} else if (NaChar <= 0.005) {
		tmp = NdChar / (Math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -3.8e+121:
		tmp = t_0
	elif NaChar <= 0.005:
		tmp = NdChar / (math.exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -3.8e+121)
		tmp = t_0;
	elseif (NaChar <= 0.005)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Vef + EDonor) + Float64(mu - Ec)) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -3.8e+121)
		tmp = t_0;
	elseif (NaChar <= 0.005)
		tmp = NdChar / (exp((((Vef + EDonor) + (mu - Ec)) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -3.8e+121], t$95$0, If[LessEqual[NaChar, 0.005], N[(NdChar / N[(N[Exp[N[(N[(N[(Vef + EDonor), $MachinePrecision] + N[(mu - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -3.8e121 or 0.0050000000000000001 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -3.8e121 < NaChar < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 72.9

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

   5. Applied rewrites 72.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -3.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 0.005:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(Vef + EDonor\right) + \left(mu - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -8.2 \cdot 10^{+169}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - -0.25 \cdot \left(mu \cdot \frac{NaChar - NdChar}{KbT}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -8.2e+169)
   (- (* 0.5 (+ NdChar NaChar)) (* -0.25 (* mu (/ (- NaChar NdChar) KbT))))
   (/ NaChar (+ (exp (/ (+ EAccept (+ Ev (- Vef mu))) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -8.2e+169) {
		tmp = (0.5 * (NdChar + NaChar)) - (-0.25 * (mu * ((NaChar - NdChar) / KbT)));
	} else {
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-8.2d+169)) then
        tmp = (0.5d0 * (ndchar + nachar)) - ((-0.25d0) * (mu * ((nachar - ndchar) / kbt)))
    else
        tmp = nachar / (exp(((eaccept + (ev + (vef - mu))) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -8.2e+169) {
		tmp = (0.5 * (NdChar + NaChar)) - (-0.25 * (mu * ((NaChar - NdChar) / KbT)));
	} else {
		tmp = NaChar / (Math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -8.2e+169:
		tmp = (0.5 * (NdChar + NaChar)) - (-0.25 * (mu * ((NaChar - NdChar) / KbT)))
	else:
		tmp = NaChar / (math.exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -8.2e+169)
		tmp = Float64(Float64(0.5 * Float64(NdChar + NaChar)) - Float64(-0.25 * Float64(mu * Float64(Float64(NaChar - NdChar) / KbT))));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev + Float64(Vef - mu))) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -8.2e+169)
		tmp = (0.5 * (NdChar + NaChar)) - (-0.25 * (mu * ((NaChar - NdChar) / KbT)));
	else
		tmp = NaChar / (exp(((EAccept + (Ev + (Vef - mu))) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -8.2e+169], N[(N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision] - N[(-0.25 * N[(mu * N[(N[(NaChar - NdChar), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -8.2000000000000006e169

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in mu around -inf

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

   8. Applied rewrites 75.0%

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

   if -8.2000000000000006e169 < 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. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 63.2

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

   5. Applied rewrites 63.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -8.2 \cdot 10^{+169}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - -0.25 \cdot \left(mu \cdot \frac{NaChar - NdChar}{KbT}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept + \left(Ev + \left(Vef - mu\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;EAccept \leq 5.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept -2.5e-59)
   (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
   (if (<= EAccept 5.2e+121)
     (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
     (/ NaChar (+ (exp (/ 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 <= -2.5e-59) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else if (EAccept <= 5.2e+121) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((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 <= (-2.5d-59)) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else if (eaccept <= 5.2d+121) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp((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 <= -2.5e-59) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else if (EAccept <= 5.2e+121) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EAccept <= -2.5e-59:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	elif EAccept <= 5.2e+121:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EAccept <= -2.5e-59)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	elseif (EAccept <= 5.2e+121)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(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 <= -2.5e-59)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	elseif (EAccept <= 5.2e+121)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EAccept, -2.5e-59], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[EAccept, 5.2e+121], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if EAccept < -2.5000000000000001e-59

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in Ev around inf

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

   8. Applied rewrites 33.0%

      \[\leadsto \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]

   if -2.5000000000000001e-59 < EAccept < 5.1999999999999998e121

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 59.6

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 49.9%

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

   if 5.1999999999999998e121 < EAccept

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 47.3%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;EAccept \leq 5.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 22.4% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -6.6 \cdot 10^{+27}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 1.05 \cdot 10^{+88}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -6.6e+27)
   (* NdChar 0.5)
   (if (<= NdChar 1.05e+88) (* NaChar 0.5) (* NdChar 0.5))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -6.6e+27) {
		tmp = NdChar * 0.5;
	} else if (NdChar <= 1.05e+88) {
		tmp = NaChar * 0.5;
	} else {
		tmp = NdChar * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-6.6d+27)) then
        tmp = ndchar * 0.5d0
    else if (ndchar <= 1.05d+88) then
        tmp = nachar * 0.5d0
    else
        tmp = ndchar * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -6.6e+27) {
		tmp = NdChar * 0.5;
	} else if (NdChar <= 1.05e+88) {
		tmp = NaChar * 0.5;
	} else {
		tmp = NdChar * 0.5;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -6.6e+27:
		tmp = NdChar * 0.5
	elif NdChar <= 1.05e+88:
		tmp = NaChar * 0.5
	else:
		tmp = NdChar * 0.5
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -6.6e+27)
		tmp = Float64(NdChar * 0.5);
	elseif (NdChar <= 1.05e+88)
		tmp = Float64(NaChar * 0.5);
	else
		tmp = Float64(NdChar * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -6.6e+27)
		tmp = NdChar * 0.5;
	elseif (NdChar <= 1.05e+88)
		tmp = NaChar * 0.5;
	else
		tmp = NdChar * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -6.6e+27], N[(NdChar * 0.5), $MachinePrecision], If[LessEqual[NdChar, 1.05e+88], N[(NaChar * 0.5), $MachinePrecision], N[(NdChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -6.5999999999999996e27 or 1.05e88 < 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 28.5

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 24.1%

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

   if -6.5999999999999996e27 < NdChar < 1.05e88

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 24.2%

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

Alternative 17: 27.1% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 26.3

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

5. Applied rewrites 26.3%

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

6. Final simplification 26.3%

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

Alternative 18: 17.4% accurate, 46.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 26.3

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

5. Applied rewrites 26.3%

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

6. Taylor expanded in NaChar around 0

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

8. Applied rewrites 16.9%

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

Reproduce

herbie shell --seed 2024222 
(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))))))