Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 14.7s

Alternatives: 22

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 2.7769419525468402e+41
  2. Ec = -7.815885387393387e-70
  3. Vef = 3.5130397960939784e-86
  4. EDonor = 2.161489113026182e-77
  5. mu = -1.80843672524981e-181
  6. KbT = -2.2143940644219216e-183
  7. NaChar = 3.544126328466806e+41
  8. Ev = 7.479387503071947e-229
  9. EAccept = 1.5691593278577118e+117

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. +-commutative N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{1 + e^{\frac{Ev + \left(Vef + \left(EAccept - mu\right)\right)}{KbT}}}, NaChar, \frac{NdChar}{1 + e^{\frac{Ec - \left(Vef + \left(mu + EDonor\right)\right)}{0 - KbT}}}\right)} \]

5. Final simplification 100.0%

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

Alternative 2: 53.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ t_2 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_3 := \frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\frac{NdChar}{t\_0} + NaChar \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-293}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-70}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_0} + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0))))
        (t_2 (+ EDonor (- (+ Vef mu) Ec)))
        (t_3 (/ NdChar (+ (exp (/ mu KbT)) 1.0))))
   (if (<= t_1 -5e-34)
     (+ (/ NdChar t_0) (* NaChar 0.5))
     (if (<= t_1 -1e-293)
       t_3
       (if (<= t_1 0.0)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_2 t_2) KbT)
             (* (- 0.0 EDonor) (fma -1.0 (/ (- Ec (+ Vef mu)) EDonor) 1.0)))
            KbT)))
         (if (<= t_1 4e-70) t_3 (+ (/ NaChar t_0) (* 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 t_0 = exp((Vef / KbT)) + 1.0;
	double t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double t_2 = EDonor + ((Vef + mu) - Ec);
	double t_3 = NdChar / (exp((mu / KbT)) + 1.0);
	double tmp;
	if (t_1 <= -5e-34) {
		tmp = (NdChar / t_0) + (NaChar * 0.5);
	} else if (t_1 <= -1e-293) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_2 * t_2) / KbT), ((0.0 - EDonor) * fma(-1.0, ((Ec - (Vef + mu)) / EDonor), 1.0))) / KbT));
	} else if (t_1 <= 4e-70) {
		tmp = t_3;
	} else {
		tmp = (NaChar / t_0) + (NdChar * 0.5);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	t_2 = Float64(EDonor + Float64(Float64(Vef + mu) - Ec))
	t_3 = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0))
	tmp = 0.0
	if (t_1 <= -5e-34)
		tmp = Float64(Float64(NdChar / t_0) + Float64(NaChar * 0.5));
	elseif (t_1 <= -1e-293)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Float64(Float64(0.0 - EDonor) * fma(-1.0, Float64(Float64(Ec - Float64(Vef + mu)) / EDonor), 1.0))) / KbT)));
	elseif (t_1 <= 4e-70)
		tmp = t_3;
	else
		tmp = Float64(Float64(NaChar / t_0) + Float64(NdChar * 0.5));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $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]}, Block[{t$95$3 = N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-34], N[(N[(NdChar / t$95$0), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-293], t$95$3, 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[(0.0 - EDonor), $MachinePrecision] * N[(-1.0 * N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / EDonor), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-70], t$95$3, N[(N[(NaChar / t$95$0), $MachinePrecision] + N[(NdChar * 0.5), $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))))) < -5.0000000000000003e-34

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 66.4

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

   5. Simplified 66.4%

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

   6. Taylor expanded in Vef around inf

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

   8. Simplified 51.3%

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

   if -5.0000000000000003e-34 < (+.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))))) < -1.0000000000000001e-293 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 3.99999999999999998e-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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 57.8

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

   5. Simplified 57.8%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 44.0

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

   8. Simplified 44.0%

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

   if -1.0000000000000001e-293 < (+.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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 96.6

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

   5. Simplified 96.6%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 87.1%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in EDonor around -inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. neg-lowering-neg.f64 N/A

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

       5. +-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. --lowering--.f64 N/A

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

       9. +-lowering-+.f64 89.0

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \color{blue}{\left(Vef + mu\right)}}{EDonor}, 1\right)\right)}{KbT}} \]

   11. Simplified 89.0%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \color{blue}{\left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)}\right)}{KbT}} \]

   if 3.99999999999999998e-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)))))

   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

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

   5. Simplified 75.4%

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

   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. *-lowering-*.f64 46.8

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

   8. Simplified 46.8%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1} + NaChar \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-293}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + NdChar \cdot 0.5\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ t_2 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_3 := \frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-293}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-70}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0))))
        (t_2 (+ EDonor (- (+ Vef mu) Ec)))
        (t_3 (/ NdChar (+ (exp (/ mu KbT)) 1.0))))
   (if (<= t_1 -5e-10)
     t_0
     (if (<= t_1 -1e-293)
       t_3
       (if (<= t_1 0.0)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_2 t_2) KbT)
             (* (- 0.0 EDonor) (fma -1.0 (/ (- Ec (+ Vef mu)) EDonor) 1.0)))
            KbT)))
         (if (<= t_1 4e-70) t_3 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 - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double t_2 = EDonor + ((Vef + mu) - Ec);
	double t_3 = NdChar / (exp((mu / KbT)) + 1.0);
	double tmp;
	if (t_1 <= -5e-10) {
		tmp = t_0;
	} else if (t_1 <= -1e-293) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_2 * t_2) / KbT), ((0.0 - EDonor) * fma(-1.0, ((Ec - (Vef + mu)) / EDonor), 1.0))) / KbT));
	} else if (t_1 <= 4e-70) {
		tmp = t_3;
	} 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(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	t_2 = Float64(EDonor + Float64(Float64(Vef + mu) - Ec))
	t_3 = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0))
	tmp = 0.0
	if (t_1 <= -5e-10)
		tmp = t_0;
	elseif (t_1 <= -1e-293)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Float64(Float64(0.0 - EDonor) * fma(-1.0, Float64(Float64(Ec - Float64(Vef + mu)) / EDonor), 1.0))) / KbT)));
	elseif (t_1 <= 4e-70)
		tmp = t_3;
	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[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $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]}, Block[{t$95$3 = N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-10], t$95$0, If[LessEqual[t$95$1, -1e-293], t$95$3, 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[(0.0 - EDonor), $MachinePrecision] * N[(-1.0 * N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / EDonor), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-70], t$95$3, 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))))) < -5.00000000000000031e-10 or 3.99999999999999998e-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)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around inf

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

   5. Simplified 74.4%

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

   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. *-lowering-*.f64 45.3

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

   8. Simplified 45.3%

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

   if -5.00000000000000031e-10 < (+.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))))) < -1.0000000000000001e-293 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 3.99999999999999998e-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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 60.8

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

   5. Simplified 60.8%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 44.3

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

   8. Simplified 44.3%

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

   if -1.0000000000000001e-293 < (+.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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 96.6

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

   5. Simplified 96.6%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 87.1%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in EDonor around -inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. neg-lowering-neg.f64 N/A

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

       5. +-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. --lowering--.f64 N/A

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

       9. +-lowering-+.f64 89.0

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \color{blue}{\left(Vef + mu\right)}}{EDonor}, 1\right)\right)}{KbT}} \]

   11. Simplified 89.0%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \color{blue}{\left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)}\right)}{KbT}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-293}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_1 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ t_3 := \frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-285}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_1 \cdot t\_1}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-70}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NaChar NdChar)))
        (t_1 (+ EDonor (- (+ Vef mu) Ec)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0))))
        (t_3 (/ NdChar (+ (exp (/ mu KbT)) 1.0))))
   (if (<= t_2 -5e-10)
     t_0
     (if (<= t_2 -5e-285)
       t_3
       (if (<= t_2 0.0)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_1 t_1) KbT)
             (* (- 0.0 EDonor) (fma -1.0 (/ (- Ec (+ Vef mu)) EDonor) 1.0)))
            KbT)))
         (if (<= t_2 4e-70) t_3 t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = EDonor + ((Vef + mu) - Ec);
	double t_2 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double t_3 = NdChar / (exp((mu / KbT)) + 1.0);
	double tmp;
	if (t_2 <= -5e-10) {
		tmp = t_0;
	} else if (t_2 <= -5e-285) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_1 * t_1) / KbT), ((0.0 - EDonor) * fma(-1.0, ((Ec - (Vef + mu)) / EDonor), 1.0))) / KbT));
	} else if (t_2 <= 4e-70) {
		tmp = t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NaChar + NdChar))
	t_1 = Float64(EDonor + Float64(Float64(Vef + mu) - Ec))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	t_3 = Float64(NdChar / Float64(exp(Float64(mu / KbT)) + 1.0))
	tmp = 0.0
	if (t_2 <= -5e-10)
		tmp = t_0;
	elseif (t_2 <= -5e-285)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_1 * t_1) / KbT), Float64(Float64(0.0 - EDonor) * fma(-1.0, Float64(Float64(Ec - Float64(Vef + mu)) / EDonor), 1.0))) / KbT)));
	elseif (t_2 <= 4e-70)
		tmp = t_3;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(EDonor + N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(NdChar / N[(N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-10], t$95$0, If[LessEqual[t$95$2, -5e-285], t$95$3, If[LessEqual[t$95$2, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(0.0 - EDonor), $MachinePrecision] * N[(-1.0 * N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / EDonor), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-70], t$95$3, 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))))) < -5.00000000000000031e-10 or 3.99999999999999998e-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)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 38.1

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

   5. Simplified 38.1%

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

   if -5.00000000000000031e-10 < (+.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))))) < -5.00000000000000018e-285 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 3.99999999999999998e-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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 61.5

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

   5. Simplified 61.5%

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

   6. Taylor expanded in mu around inf

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

      1. /-lowering-/.f64 44.8

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

   8. Simplified 44.8%

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

   if -5.00000000000000018e-285 < (+.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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 95.0

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

   5. Simplified 95.0%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 85.6%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in EDonor around -inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. neg-lowering-neg.f64 N/A

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

       5. +-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. --lowering--.f64 N/A

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

       9. +-lowering-+.f64 87.5

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \color{blue}{\left(Vef + mu\right)}}{EDonor}, 1\right)\right)}{KbT}} \]

   11. Simplified 87.5%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \color{blue}{\left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)}\right)}{KbT}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 4 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_1 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ t_3 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-285}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-282}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_1 \cdot t\_1}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 10^{-41}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NaChar NdChar)))
        (t_1 (+ EDonor (- (+ Vef mu) Ec)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0))))
        (t_3 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))))
   (if (<= t_2 -5e+134)
     t_0
     (if (<= t_2 -5e-285)
       t_3
       (if (<= t_2 2e-282)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_1 t_1) KbT)
             (* (- 0.0 EDonor) (fma -1.0 (/ (- Ec (+ Vef mu)) EDonor) 1.0)))
            KbT)))
         (if (<= t_2 1e-41) t_3 t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = EDonor + ((Vef + mu) - Ec);
	double t_2 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double t_3 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double tmp;
	if (t_2 <= -5e+134) {
		tmp = t_0;
	} else if (t_2 <= -5e-285) {
		tmp = t_3;
	} else if (t_2 <= 2e-282) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_1 * t_1) / KbT), ((0.0 - EDonor) * fma(-1.0, ((Ec - (Vef + mu)) / EDonor), 1.0))) / KbT));
	} else if (t_2 <= 1e-41) {
		tmp = t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NaChar + NdChar))
	t_1 = Float64(EDonor + Float64(Float64(Vef + mu) - Ec))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	t_3 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	tmp = 0.0
	if (t_2 <= -5e+134)
		tmp = t_0;
	elseif (t_2 <= -5e-285)
		tmp = t_3;
	elseif (t_2 <= 2e-282)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_1 * t_1) / KbT), Float64(Float64(0.0 - EDonor) * fma(-1.0, Float64(Float64(Ec - Float64(Vef + mu)) / EDonor), 1.0))) / KbT)));
	elseif (t_2 <= 1e-41)
		tmp = t_3;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(EDonor + N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+134], t$95$0, If[LessEqual[t$95$2, -5e-285], t$95$3, If[LessEqual[t$95$2, 2e-282], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(0.0 - EDonor), $MachinePrecision] * N[(-1.0 * N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / EDonor), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-41], t$95$3, 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))))) < -4.99999999999999981e134 or 1.00000000000000001e-41 < (+.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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 39.6

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

   5. Simplified 39.6%

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

   if -4.99999999999999981e134 < (+.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))))) < -5.00000000000000018e-285 or 2e-282 < (+.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))))) < 1.00000000000000001e-41

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 60.5

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

   5. Simplified 60.5%

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

   6. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 36.6

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

   8. Simplified 36.6%

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

   if -5.00000000000000018e-285 < (+.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-282

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 91.8

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

   5. Simplified 91.8%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 82.7%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in EDonor around -inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. neg-lowering-neg.f64 N/A

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

       5. +-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. --lowering--.f64 N/A

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

       9. +-lowering-+.f64 84.5

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \color{blue}{\left(Vef + mu\right)}}{EDonor}, 1\right)\right)}{KbT}} \]

   11. Simplified 84.5%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \color{blue}{\left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)}\right)}{KbT}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{+134}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-282}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 10^{-41}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_1 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ t_3 := \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_1 \cdot t\_1}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;t\_2 \leq 10^{+26}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NaChar NdChar)))
        (t_1 (+ EDonor (- (+ Vef mu) Ec)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0))))
        (t_3 (/ NaChar (+ (exp (/ Vef KbT)) 1.0))))
   (if (<= t_2 -5e-36)
     t_0
     (if (<= t_2 -2e-308)
       t_3
       (if (<= t_2 0.0)
         (/
          NdChar
          (-
           2.0
           (/
            (fma
             -0.5
             (/ (* t_1 t_1) KbT)
             (* (- 0.0 EDonor) (fma -1.0 (/ (- Ec (+ Vef mu)) EDonor) 1.0)))
            KbT)))
         (if (<= t_2 1e+26) t_3 t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = EDonor + ((Vef + mu) - Ec);
	double t_2 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double t_3 = NaChar / (exp((Vef / KbT)) + 1.0);
	double tmp;
	if (t_2 <= -5e-36) {
		tmp = t_0;
	} else if (t_2 <= -2e-308) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_1 * t_1) / KbT), ((0.0 - EDonor) * fma(-1.0, ((Ec - (Vef + mu)) / EDonor), 1.0))) / KbT));
	} else if (t_2 <= 1e+26) {
		tmp = t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NaChar + NdChar))
	t_1 = Float64(EDonor + Float64(Float64(Vef + mu) - Ec))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	t_3 = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0))
	tmp = 0.0
	if (t_2 <= -5e-36)
		tmp = t_0;
	elseif (t_2 <= -2e-308)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_1 * t_1) / KbT), Float64(Float64(0.0 - EDonor) * fma(-1.0, Float64(Float64(Ec - Float64(Vef + mu)) / EDonor), 1.0))) / KbT)));
	elseif (t_2 <= 1e+26)
		tmp = t_3;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(EDonor + N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-36], t$95$0, If[LessEqual[t$95$2, -2e-308], t$95$3, If[LessEqual[t$95$2, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(0.0 - EDonor), $MachinePrecision] * N[(-1.0 * N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / EDonor), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+26], t$95$3, 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))))) < -5.00000000000000004e-36 or 1.00000000000000005e26 < (+.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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 38.4

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

   5. Simplified 38.4%

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

   if -5.00000000000000004e-36 < (+.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))))) < -1.9999999999999998e-308 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.00000000000000005e26

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around inf

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

   5. Simplified 54.2%

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

   6. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.f64 29.8

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

   8. Simplified 29.8%

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

   if -1.9999999999999998e-308 < (+.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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 90.1%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in EDonor around -inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. neg-lowering-neg.f64 N/A

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

       5. +-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. --lowering--.f64 N/A

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

       9. +-lowering-+.f64 92.1

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \color{blue}{\left(Vef + mu\right)}}{EDonor}, 1\right)\right)}{KbT}} \]

   11. Simplified 92.1%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \color{blue}{\left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)}\right)}{KbT}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 10^{+26}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_1 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (- (+ Vef mu) Ec)))
        (t_1 (* 0.5 (+ NaChar NdChar)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0)))))
   (if (<= t_2 -1e-171)
     t_1
     (if (<= t_2 5e-145)
       (/
        NdChar
        (-
         2.0
         (/
          (fma
           -0.5
           (/ (* t_0 t_0) KbT)
           (* (- 0.0 EDonor) (fma -1.0 (/ (- Ec (+ Vef mu)) EDonor) 1.0)))
          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 * (NaChar + NdChar);
	double t_2 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -1e-171) {
		tmp = t_1;
	} else if (t_2 <= 5e-145) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_0 * t_0) / KbT), ((0.0 - EDonor) * fma(-1.0, ((Ec - (Vef + mu)) / EDonor), 1.0))) / 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(NaChar + NdChar))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -1e-171)
		tmp = t_1;
	elseif (t_2 <= 5e-145)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_0 * t_0) / KbT), Float64(Float64(0.0 - EDonor) * fma(-1.0, Float64(Float64(Ec - Float64(Vef + mu)) / EDonor), 1.0))) / 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[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-171], t$95$1, If[LessEqual[t$95$2, 5e-145], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(0.0 - EDonor), $MachinePrecision] * N[(-1.0 * N[(N[(Ec - N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / EDonor), $MachinePrecision] + 1.0), $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))))) < -9.9999999999999998e-172 or 4.9999999999999998e-145 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 33.6

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

   5. Simplified 33.6%

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

   if -9.9999999999999998e-172 < (+.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))))) < 4.9999999999999998e-145

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 82.7

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

   5. Simplified 82.7%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 61.4%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in EDonor around -inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. neg-lowering-neg.f64 N/A

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

       5. +-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. --lowering--.f64 N/A

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

       9. +-lowering-+.f64 62.1

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \color{blue}{\left(Vef + mu\right)}}{EDonor}, 1\right)\right)}{KbT}} \]

   11. Simplified 62.1%

       \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \color{blue}{\left(-EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)}\right)}{KbT}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(0 - EDonor\right) \cdot \mathsf{fma}\left(-1, \frac{Ec - \left(Vef + mu\right)}{EDonor}, 1\right)\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(\left(Vef + mu\right) - Ec\right)\\ t_1 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_2 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_0 \cdot t\_0}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (- (+ Vef mu) Ec)))
        (t_1 (* 0.5 (+ NaChar NdChar)))
        (t_2
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0)))))
   (if (<= t_2 -1e-171)
     t_1
     (if (<= t_2 5e-145)
       (/
        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 * (NaChar + NdChar);
	double t_2 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_2 <= -1e-171) {
		tmp = t_1;
	} else if (t_2 <= 5e-145) {
		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(NaChar + NdChar))
	t_2 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -1e-171)
		tmp = t_1;
	elseif (t_2 <= 5e-145)
		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[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-171], t$95$1, If[LessEqual[t$95$2, 5e-145], 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))))) < -9.9999999999999998e-172 or 4.9999999999999998e-145 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 33.6

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

   5. Simplified 33.6%

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

   if -9.9999999999999998e-172 < (+.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))))) < 4.9999999999999998e-145

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 82.7

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

   5. Simplified 82.7%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 61.4%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, \left(Ec - \left(Vef + mu\right)\right) - EDonor\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ t_1 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_2 := Vef + \left(mu - Ec\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(Vef + mu\right) - \mathsf{fma}\left(-0.5, \frac{t\_2 \cdot t\_2}{KbT}, Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0))))
        (t_1 (* 0.5 (+ NaChar NdChar)))
        (t_2 (+ Vef (- mu Ec))))
   (if (<= t_0 -1e-171)
     t_1
     (if (<= t_0 5e-145)
       (/
        NdChar
        (+ 2.0 (/ (- (+ Vef mu) (fma -0.5 (/ (* t_2 t_2) KbT) Ec)) 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 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double t_1 = 0.5 * (NaChar + NdChar);
	double t_2 = Vef + (mu - Ec);
	double tmp;
	if (t_0 <= -1e-171) {
		tmp = t_1;
	} else if (t_0 <= 5e-145) {
		tmp = NdChar / (2.0 + (((Vef + mu) - fma(-0.5, ((t_2 * t_2) / KbT), Ec)) / KbT));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	t_1 = Float64(0.5 * Float64(NaChar + NdChar))
	t_2 = Float64(Vef + Float64(mu - Ec))
	tmp = 0.0
	if (t_0 <= -1e-171)
		tmp = t_1;
	elseif (t_0 <= 5e-145)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Vef + mu) - fma(-0.5, Float64(Float64(t_2 * t_2) / KbT), Ec)) / 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[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-171], t$95$1, If[LessEqual[t$95$0, 5e-145], N[(NdChar / N[(2.0 + N[(N[(N[(Vef + mu), $MachinePrecision] - N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / KbT), $MachinePrecision] + Ec), $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))))) < -9.9999999999999998e-172 or 4.9999999999999998e-145 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 33.6

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

   5. Simplified 33.6%

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

   if -9.9999999999999998e-172 < (+.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))))) < 4.9999999999999998e-145

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 82.7

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

   5. Simplified 82.7%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 61.4%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in EDonor around 0

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

       1. --lowering--.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. unpow2 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(\left(Vef + mu\right) - Ec\right) \cdot \left(\left(Vef + mu\right) - Ec\right)}}{KbT}, Ec\right) - \left(Vef + mu\right)}{KbT}} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(\left(Vef + mu\right) - Ec\right) \cdot \left(\left(Vef + mu\right) - Ec\right)}}{KbT}, Ec\right) - \left(Vef + mu\right)}{KbT}} \]

       7. associate--l+ N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(Vef + \left(mu - Ec\right)\right)} \cdot \left(\left(Vef + mu\right) - Ec\right)}{KbT}, Ec\right) - \left(Vef + mu\right)}{KbT}} \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(Vef + \left(mu - Ec\right)\right)} \cdot \left(\left(Vef + mu\right) - Ec\right)}{KbT}, Ec\right) - \left(Vef + mu\right)}{KbT}} \]

       9. --lowering--.f64 N/A

          \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(Vef + \color{blue}{\left(mu - Ec\right)}\right) \cdot \left(\left(Vef + mu\right) - Ec\right)}{KbT}, Ec\right) - \left(Vef + mu\right)}{KbT}} \]

       10. associate--l+ N/A

           \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(Vef + \left(mu - Ec\right)\right) \cdot \color{blue}{\left(Vef + \left(mu - Ec\right)\right)}}{KbT}, Ec\right) - \left(Vef + mu\right)}{KbT}} \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(Vef + \left(mu - Ec\right)\right) \cdot \color{blue}{\left(Vef + \left(mu - Ec\right)\right)}}{KbT}, Ec\right) - \left(Vef + mu\right)}{KbT}} \]

       12. --lowering--.f64 N/A

           \[\leadsto \frac{NdChar}{2 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(Vef + \left(mu - Ec\right)\right) \cdot \left(Vef + \color{blue}{\left(mu - Ec\right)}\right)}{KbT}, Ec\right) - \left(Vef + mu\right)}{KbT}} \]

       13. +-lowering-+.f64 57.5

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

   11. Simplified 57.5%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{\left(Vef + mu\right) - \mathsf{fma}\left(-0.5, \frac{\left(Vef + \left(mu - Ec\right)\right) \cdot \left(Vef + \left(mu - Ec\right)\right)}{KbT}, Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-268}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-274}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(Vef \cdot Vef\right)}{KbT \cdot KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NaChar NdChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-268)
     t_0
     (if (<= t_1 2e-274) (/ NdChar (/ (* 0.5 (* Vef Vef)) (* KbT KbT))) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-268) {
		tmp = t_0;
	} else if (t_1 <= 2e-274) {
		tmp = NdChar / ((0.5 * (Vef * Vef)) / (KbT * KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (nachar + ndchar)
    t_1 = (ndchar / (exp(((mu + (edonor - (ec - vef))) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-2d-268)) then
        tmp = t_0
    else if (t_1 <= 2d-274) then
        tmp = ndchar / ((0.5d0 * (vef * vef)) / (kbt * kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-268) {
		tmp = t_0;
	} else if (t_1 <= 2e-274) {
		tmp = NdChar / ((0.5 * (Vef * Vef)) / (KbT * KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NaChar + NdChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -2e-268:
		tmp = t_0
	elif t_1 <= 2e-274:
		tmp = NdChar / ((0.5 * (Vef * Vef)) / (KbT * KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NaChar + NdChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-268)
		tmp = t_0;
	elseif (t_1 <= 2e-274)
		tmp = Float64(NdChar / Float64(Float64(0.5 * Float64(Vef * Vef)) / Float64(KbT * KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NaChar + NdChar);
	t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-268)
		tmp = t_0;
	elseif (t_1 <= 2e-274)
		tmp = NdChar / ((0.5 * (Vef * Vef)) / (KbT * KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-268], t$95$0, If[LessEqual[t$95$1, 2e-274], N[(NdChar / N[(N[(0.5 * N[(Vef * Vef), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.99999999999999992e-268 or 1.99999999999999993e-274 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 31.9

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

   5. Simplified 31.9%

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

   if -1.99999999999999992e-268 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.99999999999999993e-274

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 91.1

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

   5. Simplified 91.1%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 75.3%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in Vef around inf

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

          \[\leadsto \frac{NdChar}{\frac{\frac{1}{2} \cdot \color{blue}{\left(Vef \cdot Vef\right)}}{{KbT}^{2}}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{NdChar}{\frac{\frac{1}{2} \cdot \color{blue}{\left(Vef \cdot Vef\right)}}{{KbT}^{2}}} \]

       6. unpow2 N/A

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

       7. *-lowering-*.f64 46.7

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

   11. Simplified 46.7%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-274}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(Vef \cdot Vef\right)}{KbT \cdot KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-268}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-274}:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Vef \cdot Vef}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NaChar NdChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-268)
     t_0
     (if (<= t_1 2e-274) (/ (* 2.0 (* NdChar (* KbT KbT))) (* Vef Vef)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-268) {
		tmp = t_0;
	} else if (t_1 <= 2e-274) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef);
	} 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 * (nachar + ndchar)
    t_1 = (ndchar / (exp(((mu + (edonor - (ec - vef))) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-2d-268)) then
        tmp = t_0
    else if (t_1 <= 2d-274) then
        tmp = (2.0d0 * (ndchar * (kbt * kbt))) / (vef * vef)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-268) {
		tmp = t_0;
	} else if (t_1 <= 2e-274) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NaChar + NdChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -2e-268:
		tmp = t_0
	elif t_1 <= 2e-274:
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NaChar + NdChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-268)
		tmp = t_0;
	elseif (t_1 <= 2e-274)
		tmp = Float64(Float64(2.0 * Float64(NdChar * Float64(KbT * KbT))) / Float64(Vef * Vef));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NaChar + NdChar);
	t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-268)
		tmp = t_0;
	elseif (t_1 <= 2e-274)
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Vef * Vef);
	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[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-268], t$95$0, If[LessEqual[t$95$1, 2e-274], N[(N[(2.0 * N[(NdChar * N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Vef * Vef), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.99999999999999992e-268 or 1.99999999999999993e-274 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 31.9

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

   5. Simplified 31.9%

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

   if -1.99999999999999992e-268 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.99999999999999993e-274

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 91.1

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

   5. Simplified 91.1%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 75.3%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in Vef around inf

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

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}{{Vef}^{2}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}{{Vef}^{2}}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}}{{Vef}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \frac{2 \cdot \color{blue}{\left(NdChar \cdot {KbT}^{2}\right)}}{{Vef}^{2}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \color{blue}{\left(NdChar \cdot {KbT}^{2}\right)}}{{Vef}^{2}} \]

       6. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(NdChar \cdot \color{blue}{\left(KbT \cdot KbT\right)}\right)}{{Vef}^{2}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(NdChar \cdot \color{blue}{\left(KbT \cdot KbT\right)}\right)}{{Vef}^{2}} \]

       8. unpow2 N/A

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

       9. *-lowering-*.f64 40.2

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

   11. Simplified 40.2%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 2 \cdot 10^{-274}:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Vef \cdot Vef}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-268}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Ec \cdot Ec}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NaChar NdChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-268)
     t_0
     (if (<= t_1 5e-293) (/ (* 2.0 (* NdChar (* KbT KbT))) (* Ec Ec)) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-268) {
		tmp = t_0;
	} else if (t_1 <= 5e-293) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec);
	} 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 * (nachar + ndchar)
    t_1 = (ndchar / (exp(((mu + (edonor - (ec - vef))) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-2d-268)) then
        tmp = t_0
    else if (t_1 <= 5d-293) then
        tmp = (2.0d0 * (ndchar * (kbt * kbt))) / (ec * ec)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-268) {
		tmp = t_0;
	} else if (t_1 <= 5e-293) {
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NaChar + NdChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -2e-268:
		tmp = t_0
	elif t_1 <= 5e-293:
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NaChar + NdChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-268)
		tmp = t_0;
	elseif (t_1 <= 5e-293)
		tmp = Float64(Float64(2.0 * Float64(NdChar * Float64(KbT * KbT))) / Float64(Ec * Ec));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NaChar + NdChar);
	t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-268)
		tmp = t_0;
	elseif (t_1 <= 5e-293)
		tmp = (2.0 * (NdChar * (KbT * KbT))) / (Ec * Ec);
	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[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-268], t$95$0, If[LessEqual[t$95$1, 5e-293], N[(N[(2.0 * N[(NdChar * N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Ec * Ec), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.99999999999999992e-268 or 5.0000000000000003e-293 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 31.6

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

   5. Simplified 31.6%

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

   if -1.99999999999999992e-268 < (+.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))))) < 5.0000000000000003e-293

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 92.3

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

   5. Simplified 92.3%

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

   6. Taylor expanded in KbT around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 77.7%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right) \cdot \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)}{KbT}, 0 - \left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right)\right)}{KbT}}} \]

   9. Taylor expanded in Ec around inf

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

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}{{Ec}^{2}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}{{Ec}^{2}}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left({KbT}^{2} \cdot NdChar\right)}}{{Ec}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \frac{2 \cdot \color{blue}{\left(NdChar \cdot {KbT}^{2}\right)}}{{Ec}^{2}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \color{blue}{\left(NdChar \cdot {KbT}^{2}\right)}}{{Ec}^{2}} \]

       6. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(NdChar \cdot \color{blue}{\left(KbT \cdot KbT\right)}\right)}{{Ec}^{2}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(NdChar \cdot \color{blue}{\left(KbT \cdot KbT\right)}\right)}{{Ec}^{2}} \]

       8. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{\color{blue}{Ec \cdot Ec}} \]

       9. *-lowering-*.f64 35.6

          \[\leadsto \frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{\color{blue}{Ec \cdot Ec}} \]

   11. Simplified 35.6%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\frac{2 \cdot \left(NdChar \cdot \left(KbT \cdot KbT\right)\right)}{Ec \cdot Ec}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NaChar NdChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0)))))
   (if (<= t_1 -1e-132)
     t_0
     (if (<= t_1 5e-145) (/ NdChar (+ 2.0 (/ EDonor KbT))) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-132) {
		tmp = t_0;
	} else if (t_1 <= 5e-145) {
		tmp = NdChar / (2.0 + (EDonor / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (nachar + ndchar)
    t_1 = (ndchar / (exp(((mu + (edonor - (ec - vef))) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-1d-132)) then
        tmp = t_0
    else if (t_1 <= 5d-145) then
        tmp = ndchar / (2.0d0 + (edonor / kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -1e-132) {
		tmp = t_0;
	} else if (t_1 <= 5e-145) {
		tmp = NdChar / (2.0 + (EDonor / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NaChar + NdChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -1e-132:
		tmp = t_0
	elif t_1 <= 5e-145:
		tmp = NdChar / (2.0 + (EDonor / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NaChar + NdChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-132)
		tmp = t_0;
	elseif (t_1 <= 5e-145)
		tmp = Float64(NdChar / Float64(2.0 + Float64(EDonor / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NaChar + NdChar);
	t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -1e-132)
		tmp = t_0;
	elseif (t_1 <= 5e-145)
		tmp = NdChar / (2.0 + (EDonor / KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-132], t$95$0, If[LessEqual[t$95$1, 5e-145], N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999999e-133 or 4.9999999999999998e-145 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 34.5

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

   5. Simplified 34.5%

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

   if -9.9999999999999999e-133 < (+.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))))) < 4.9999999999999998e-145

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 80.6

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

   5. Simplified 80.6%

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

   6. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 40.1

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

   8. Simplified 40.1%

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

   9. Taylor expanded in EDonor around 0

      \[\leadsto \frac{NdChar}{\color{blue}{2 + \frac{EDonor}{KbT}}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{2 + \frac{EDonor}{KbT}}} \]

       2. /-lowering-/.f64 27.6

          \[\leadsto \frac{NdChar}{2 + \color{blue}{\frac{EDonor}{KbT}}} \]

   11. Simplified 27.6%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -1 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NaChar + NdChar\right)\\ t_1 := \frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-269}:\\ \;\;\;\;\frac{0.25 \cdot \left(NdChar \cdot Ec\right)}{KbT}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NaChar NdChar)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) 1.0)))))
   (if (<= t_1 -2e-308)
     t_0
     (if (<= t_1 1e-269) (/ (* 0.25 (* NdChar Ec)) KbT) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-308) {
		tmp = t_0;
	} else if (t_1 <= 1e-269) {
		tmp = (0.25 * (NdChar * Ec)) / KbT;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (nachar + ndchar)
    t_1 = (ndchar / (exp(((mu + (edonor - (ec - vef))) / kbt)) + 1.0d0)) + (nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) + 1.0d0))
    if (t_1 <= (-2d-308)) then
        tmp = t_0
    else if (t_1 <= 1d-269) then
        tmp = (0.25d0 * (ndchar * ec)) / kbt
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NaChar + NdChar);
	double t_1 = (NdChar / (Math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_1 <= -2e-308) {
		tmp = t_0;
	} else if (t_1 <= 1e-269) {
		tmp = (0.25 * (NdChar * Ec)) / KbT;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NaChar + NdChar)
	t_1 = (NdChar / (math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0))
	tmp = 0
	if t_1 <= -2e-308:
		tmp = t_0
	elif t_1 <= 1e-269:
		tmp = (0.25 * (NdChar * Ec)) / KbT
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NaChar + NdChar))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-308)
		tmp = t_0;
	elseif (t_1 <= 1e-269)
		tmp = Float64(Float64(0.25 * Float64(NdChar * Ec)) / KbT);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NaChar + NdChar);
	t_1 = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-308)
		tmp = t_0;
	elseif (t_1 <= 1e-269)
		tmp = (0.25 * (NdChar * Ec)) / KbT;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-308], t$95$0, If[LessEqual[t$95$1, 1e-269], N[(N[(0.25 * N[(NdChar * Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.9999999999999998e-308 or 9.9999999999999996e-270 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 31.0

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

   5. Simplified 31.0%

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

   if -1.9999999999999998e-308 < (+.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-270

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 4.8

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

   5. Simplified 4.8%

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

   6. Taylor expanded in KbT around inf

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-lowering-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. +-lowering-+.f64 1.9

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

   8. Simplified 1.9%

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

   9. Taylor expanded in Ec around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{Ec \cdot NdChar}{KbT}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left(Ec \cdot NdChar\right)}{KbT}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left(Ec \cdot NdChar\right)}{KbT}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \left(Ec \cdot NdChar\right)}}{KbT} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(NdChar \cdot Ec\right)}}{KbT} \]

       5. *-lowering-*.f64 22.8

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

   11. Simplified 22.8%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1} \leq 10^{-269}:\\ \;\;\;\;\frac{0.25 \cdot \left(NdChar \cdot Ec\right)}{KbT}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -1.75 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 4.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)) 1.0))))
   (if (<= NaChar -1.75e+43)
     t_0
     (if (<= NaChar 4.4e+145)
       (+
        (/ NdChar (+ (exp (/ (+ mu (- EDonor (- Ec Vef))) KbT)) 1.0))
        (/ NaChar (+ (exp (/ Vef KbT)) 1.0)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.75e+43) {
		tmp = t_0;
	} else if (NaChar <= 4.4e+145) {
		tmp = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((Vef / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp((((ev + eaccept) + (vef - mu)) / kbt)) + 1.0d0)
    if (nachar <= (-1.75d+43)) then
        tmp = t_0
    else if (nachar <= 4.4d+145) then
        tmp = (ndchar / (exp(((mu + (edonor - (ec - vef))) / kbt)) + 1.0d0)) + (nachar / (exp((vef / kbt)) + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.75e+43) {
		tmp = t_0;
	} else if (NaChar <= 4.4e+145) {
		tmp = (NdChar / (Math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (Math.exp((Vef / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -1.75e+43:
		tmp = t_0
	elif NaChar <= 4.4e+145:
		tmp = (NdChar / (math.exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (math.exp((Vef / KbT)) + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -1.75e+43)
		tmp = t_0;
	elseif (NaChar <= 4.4e+145)
		tmp = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor - Float64(Ec - Vef))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -1.75e+43)
		tmp = t_0;
	elseif (NaChar <= 4.4e+145)
		tmp = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((Vef / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.75e+43], t$95$0, If[LessEqual[NaChar, 4.4e+145], N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor - N[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.7500000000000001e43 or 4.40000000000000017e145 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 78.8

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

   5. Simplified 78.8%

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

   if -1.7500000000000001e43 < NaChar < 4.40000000000000017e145

   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

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

   5. Simplified 73.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.75 \cdot 10^{+43}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 4.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{mu + \left(EDonor - \left(Ec - Vef\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / (exp(((mu + (EDonor - (Ec - Vef))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Ev + Vef)) - 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[(Ec - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $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%

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

Alternative 17: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -1.85 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(\left(Vef + mu\right) - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)) 1.0))))
   (if (<= NaChar -1.85e+62)
     t_0
     (if (<= NaChar 5.2e+67)
       (/ NdChar (+ (exp (/ (+ EDonor (- (+ Vef 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((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.85e+62) {
		tmp = t_0;
	} else if (NaChar <= 5.2e+67) {
		tmp = NdChar / (exp(((EDonor + ((Vef + 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((((ev + eaccept) + (vef - mu)) / kbt)) + 1.0d0)
    if (nachar <= (-1.85d+62)) then
        tmp = t_0
    else if (nachar <= 5.2d+67) then
        tmp = ndchar / (exp(((edonor + ((vef + 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((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.85e+62) {
		tmp = t_0;
	} else if (NaChar <= 5.2e+67) {
		tmp = NdChar / (Math.exp(((EDonor + ((Vef + 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((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -1.85e+62:
		tmp = t_0
	elif NaChar <= 5.2e+67:
		tmp = NdChar / (math.exp(((EDonor + ((Vef + 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(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -1.85e+62)
		tmp = t_0;
	elseif (NaChar <= 5.2e+67)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Float64(Vef + 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((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -1.85e+62)
		tmp = t_0;
	elseif (NaChar <= 5.2e+67)
		tmp = NdChar / (exp(((EDonor + ((Vef + 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[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.85e+62], t$95$0, If[LessEqual[NaChar, 5.2e+67], N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.85000000000000007e62 or 5.2000000000000001e67 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 73.8

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

   5. Simplified 73.8%

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

   if -1.85000000000000007e62 < NaChar < 5.2000000000000001e67

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 71.7

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

   5. Simplified 71.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.85 \cdot 10^{+62}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(\left(Vef + mu\right) - Ec\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ t_1 := \frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{if}\;EDonor \leq -2.25 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;EDonor \leq 1.12 \cdot 10^{-48}:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{elif}\;EDonor \leq 3.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0))
        (t_1 (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))))
   (if (<= EDonor -2.25e+61)
     t_1
     (if (<= EDonor 1.12e-48)
       (/ NaChar t_0)
       (if (<= EDonor 3.9e+100) (/ NdChar t_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 = exp((Vef / KbT)) + 1.0;
	double t_1 = NdChar / (exp((EDonor / KbT)) + 1.0);
	double tmp;
	if (EDonor <= -2.25e+61) {
		tmp = t_1;
	} else if (EDonor <= 1.12e-48) {
		tmp = NaChar / t_0;
	} else if (EDonor <= 3.9e+100) {
		tmp = NdChar / t_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) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    t_1 = ndchar / (exp((edonor / kbt)) + 1.0d0)
    if (edonor <= (-2.25d+61)) then
        tmp = t_1
    else if (edonor <= 1.12d-48) then
        tmp = nachar / t_0
    else if (edonor <= 3.9d+100) then
        tmp = ndchar / t_0
    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 = Math.exp((Vef / KbT)) + 1.0;
	double t_1 = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	double tmp;
	if (EDonor <= -2.25e+61) {
		tmp = t_1;
	} else if (EDonor <= 1.12e-48) {
		tmp = NaChar / t_0;
	} else if (EDonor <= 3.9e+100) {
		tmp = NdChar / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	t_1 = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	tmp = 0
	if EDonor <= -2.25e+61:
		tmp = t_1
	elif EDonor <= 1.12e-48:
		tmp = NaChar / t_0
	elif EDonor <= 3.9e+100:
		tmp = NdChar / t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	t_1 = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0))
	tmp = 0.0
	if (EDonor <= -2.25e+61)
		tmp = t_1;
	elseif (EDonor <= 1.12e-48)
		tmp = Float64(NaChar / t_0);
	elseif (EDonor <= 3.9e+100)
		tmp = Float64(NdChar / t_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 = exp((Vef / KbT)) + 1.0;
	t_1 = NdChar / (exp((EDonor / KbT)) + 1.0);
	tmp = 0.0;
	if (EDonor <= -2.25e+61)
		tmp = t_1;
	elseif (EDonor <= 1.12e-48)
		tmp = NaChar / t_0;
	elseif (EDonor <= 3.9e+100)
		tmp = NdChar / t_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[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EDonor, -2.25e+61], t$95$1, If[LessEqual[EDonor, 1.12e-48], N[(NaChar / t$95$0), $MachinePrecision], If[LessEqual[EDonor, 3.9e+100], N[(NdChar / t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if EDonor < -2.25e61 or 3.9e100 < EDonor

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 65.7

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

   5. Simplified 65.7%

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

   6. Taylor expanded in EDonor around inf

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

      1. /-lowering-/.f64 50.8

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

   8. Simplified 50.8%

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

   if -2.25e61 < EDonor < 1.11999999999999999e-48

   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

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

   5. Simplified 67.0%

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

   6. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.f64 42.9

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

   8. Simplified 42.9%

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

   if 1.11999999999999999e-48 < EDonor < 3.9e100

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.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. +-lowering-+.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. +-lowering-+.f64 61.1

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

   5. Simplified 61.1%

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

   6. Taylor expanded in Vef around inf

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

      1. /-lowering-/.f64 47.1

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

   8. Simplified 47.1%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -2.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 1.12 \cdot 10^{-48}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;EDonor \leq 3.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -7.8 \cdot 10^{+261}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -7.8e+261)
   (* 0.5 (+ NaChar NdChar))
   (/ NaChar (+ (exp (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -7.8e+261) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-7.8d+261)) then
        tmp = 0.5d0 * (nachar + ndchar)
    else
        tmp = nachar / (exp((((ev + eaccept) + (vef - mu)) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -7.8e+261) {
		tmp = 0.5 * (NaChar + NdChar);
	} else {
		tmp = NaChar / (Math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -7.8e+261:
		tmp = 0.5 * (NaChar + NdChar)
	else:
		tmp = NaChar / (math.exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -7.8e+261)
		tmp = Float64(0.5 * Float64(NaChar + NdChar));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Ev + EAccept) + 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 <= -7.8e+261)
		tmp = 0.5 * (NaChar + NdChar);
	else
		tmp = NaChar / (exp((((Ev + EAccept) + (Vef - mu)) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -7.8e+261], N[(0.5 * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -7.79999999999999988e261

   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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 93.7

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

   5. Simplified 93.7%

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

   if -7.79999999999999988e261 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. --lowering--.f64 62.3

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

   5. Simplified 62.3%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -7.8 \cdot 10^{+261}:\\ \;\;\;\;0.5 \cdot \left(NaChar + NdChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 21.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -5.4 \cdot 10^{-141}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq 6 \cdot 10^{+132}:\\ \;\;\;\;NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -5.4e-141)
   (* NaChar 0.5)
   (if (<= NaChar 6e+132) (* NdChar 0.5) (* NaChar 0.5))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -5.4e-141) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 6e+132) {
		tmp = NdChar * 0.5;
	} else {
		tmp = NaChar * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-5.4d-141)) then
        tmp = nachar * 0.5d0
    else if (nachar <= 6d+132) then
        tmp = ndchar * 0.5d0
    else
        tmp = nachar * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -5.4e-141) {
		tmp = NaChar * 0.5;
	} else if (NaChar <= 6e+132) {
		tmp = NdChar * 0.5;
	} else {
		tmp = NaChar * 0.5;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -5.4e-141:
		tmp = NaChar * 0.5
	elif NaChar <= 6e+132:
		tmp = NdChar * 0.5
	else:
		tmp = NaChar * 0.5
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -5.4e-141)
		tmp = Float64(NaChar * 0.5);
	elseif (NaChar <= 6e+132)
		tmp = Float64(NdChar * 0.5);
	else
		tmp = Float64(NaChar * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -5.4e-141)
		tmp = NaChar * 0.5;
	elseif (NaChar <= 6e+132)
		tmp = NdChar * 0.5;
	else
		tmp = NaChar * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -5.4e-141], N[(NaChar * 0.5), $MachinePrecision], If[LessEqual[NaChar, 6e+132], N[(NdChar * 0.5), $MachinePrecision], N[(NaChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -5.4000000000000005e-141 or 5.9999999999999996e132 < NaChar

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 24.9

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

   5. Simplified 24.9%

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

   6. Taylor expanded in NaChar around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 21.6

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

   8. Simplified 21.6%

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

   if -5.4000000000000005e-141 < NaChar < 5.9999999999999996e132

   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. *-lowering-*.f64 N/A

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

      3. +-lowering-+.f64 25.4

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

   5. Simplified 25.4%

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

   6. Taylor expanded in NaChar around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 22.2

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

   8. Simplified 22.2%

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

Alternative 21: 27.3% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-lowering-+.f64 25.1

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

5. Simplified 25.1%

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

Alternative 22: 18.1% accurate, 46.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-lowering-+.f64 25.1

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

5. Simplified 25.1%

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

6. Taylor expanded in NaChar around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 17.3

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

8. Simplified 17.3%

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

Reproduce

herbie shell --seed 2024196 
(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))))))