Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 15.8s

Alternatives: 22

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 1.307094054717946e+133
  2. Ec = 5.091790416387511e-86
  3. Vef = 6.685850239258612e+267
  4. EDonor = 7.826467230273407e-176
  5. mu = -1.8818126465041044e-95
  6. KbT = -3.87520422930828e+134
  7. NaChar = -3.4639398184349867e+132
  8. Ev = -3.2240085232216155e+97
  9. EAccept = 2.5587401263449107e+246

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + {\left(\mathsf{E}\left(\right) \cdot \sqrt{\mathsf{E}\left(\right)}\right)}^{\left(2 \cdot \left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) \cdot \frac{\frac{0.5}{KbT}}{2}\right)\right)} \cdot e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT} \cdot 0.25}} \end{array} \]

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
    (*
     (pow
      (* (E) (sqrt (E)))
      (* 2.0 (* (- (+ EAccept (+ Ev Vef)) mu) (/ (/ 0.5 KbT) 2.0))))
     (exp (* (/ (- (+ (+ Vef Ev) EAccept) mu) KbT) 0.25)))))))

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 100.0

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

   10. lift-+.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 100.0

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. sub-neg N/A

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

   17. lift-neg.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. associate-/l/ N/A

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

   21. lower-/.f64 N/A

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{{\left(\mathsf{E}\left(\right) \cdot \sqrt{\mathsf{E}\left(\right)}\right)}^{\left(2 \cdot \left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) \cdot \frac{\frac{0.5}{KbT}}{2}\right)\right)} \cdot {\left(e^{\frac{0.5}{KbT} \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}\right)}^{0.5}}} \]

10. Applied rewrites 100.0%

    \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + {\left(\mathsf{E}\left(\right) \cdot \sqrt{\mathsf{E}\left(\right)}\right)}^{\left(2 \cdot \left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right) \cdot \frac{\frac{0.5}{KbT}}{2}\right)\right)} \cdot \color{blue}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) - mu}{KbT} \cdot 0.25}}} \]
11. Add Preprocessing

Alternative 2: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-226}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;t\_1 \leq 10^{-64}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          t_0)))
   (if (<= t_1 -4e-226)
     (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) t_0)
     (if (<= t_1 5e-302)
       (/ NdChar (+ (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) 1.0))
       (if (<= t_1 1e-64)
         (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_0)
         (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0;
	double tmp;
	if (t_1 <= -4e-226) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + t_0;
	} else if (t_1 <= 5e-302) {
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else if (t_1 <= 1e-64) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	} else {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + t_0
    if (t_1 <= (-4d-226)) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + t_0
    else if (t_1 <= 5d-302) then
        tmp = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) + 1.0d0)
    else if (t_1 <= 1d-64) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + t_0
    else
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + 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 / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0;
	double tmp;
	if (t_1 <= -4e-226) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + t_0;
	} else if (t_1 <= 5e-302) {
		tmp = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else if (t_1 <= 1e-64) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + t_0;
	} else {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0
	tmp = 0
	if t_1 <= -4e-226:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + t_0
	elif t_1 <= 5e-302:
		tmp = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0)
	elif t_1 <= 1e-64:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + t_0
	else:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + t_0)
	tmp = 0.0
	if (t_1 <= -4e-226)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + t_0);
	elseif (t_1 <= 5e-302)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) + 1.0));
	elseif (t_1 <= 1e-64)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + t_0);
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0;
	tmp = 0.0;
	if (t_1 <= -4e-226)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + t_0;
	elseif (t_1 <= 5e-302)
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	elseif (t_1 <= 1e-64)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	else
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + 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[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-226], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e-302], N[(NdChar / N[(N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-64], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $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))))) < -3.99999999999999969e-226

   1. Initial program 100.0%

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

   3. Taylor expanded in mu around inf

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

      1. lower-/.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -3.99999999999999969e-226 < (+.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))))) < 5.00000000000000033e-302

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 16.3%

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

   9. Taylor expanded in NdChar around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-+.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 95.3

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

   11. Applied rewrites 95.3%

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

   if 5.00000000000000033e-302 < (+.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))))) < 9.99999999999999965e-65

   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 EDonor around inf

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

      1. lower-/.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if 9.99999999999999965e-65 < (+.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

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

      1. lower-/.f64 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -4 \cdot 10^{-226}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 10^{-64}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + t\_0\\ t_2 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + t\_0\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-226}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;t\_2 \leq 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))
        (t_1 (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_0))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          t_0)))
   (if (<= t_2 -4e-226)
     t_1
     (if (<= t_2 5e-302)
       (/ NdChar (+ (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) 1.0))
       (if (<= t_2 1e-64) t_1 (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	double t_2 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0;
	double tmp;
	if (t_2 <= -4e-226) {
		tmp = t_1;
	} else if (t_2 <= 5e-302) {
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else if (t_2 <= 1e-64) {
		tmp = t_1;
	} else {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + 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) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((edonor / kbt)))) + t_0
    t_2 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + t_0
    if (t_2 <= (-4d-226)) then
        tmp = t_1
    else if (t_2 <= 5d-302) then
        tmp = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) + 1.0d0)
    else if (t_2 <= 1d-64) then
        tmp = t_1
    else
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + 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 / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + t_0;
	double t_2 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0;
	double tmp;
	if (t_2 <= -4e-226) {
		tmp = t_1;
	} else if (t_2 <= 5e-302) {
		tmp = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else if (t_2 <= 1e-64) {
		tmp = t_1;
	} else {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + t_0
	t_2 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0
	tmp = 0
	if t_2 <= -4e-226:
		tmp = t_1
	elif t_2 <= 5e-302:
		tmp = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0)
	elif t_2 <= 1e-64:
		tmp = t_1
	else:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + t_0)
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + t_0)
	tmp = 0.0
	if (t_2 <= -4e-226)
		tmp = t_1;
	elseif (t_2 <= 5e-302)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) + 1.0));
	elseif (t_2 <= 1e-64)
		tmp = t_1;
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	t_1 = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	t_2 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0;
	tmp = 0.0;
	if (t_2 <= -4e-226)
		tmp = t_1;
	elseif (t_2 <= 5e-302)
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	elseif (t_2 <= 1e-64)
		tmp = t_1;
	else
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + 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[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-226], t$95$1, If[LessEqual[t$95$2, 5e-302], N[(NdChar / N[(N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-64], t$95$1, N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -3.99999999999999969e-226 or 5.00000000000000033e-302 < (+.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))))) < 9.99999999999999965e-65

   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 EDonor around inf

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

      1. lower-/.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if -3.99999999999999969e-226 < (+.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))))) < 5.00000000000000033e-302

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 16.3%

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

   9. Taylor expanded in NdChar around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-+.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 95.3

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

   11. Applied rewrites 95.3%

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

   if 9.99999999999999965e-65 < (+.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

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

      1. lower-/.f64 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -4 \cdot 10^{-226}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-302}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 10^{-64}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-226} \lor \neg \left(t\_1 \leq 5 \cdot 10^{-302}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          t_0)))
   (if (or (<= t_1 -4e-226) (not (<= t_1 5e-302)))
     (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_0)
     (/ NdChar (+ (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0;
	double tmp;
	if ((t_1 <= -4e-226) || !(t_1 <= 5e-302)) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	} else {
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + t_0
    if ((t_1 <= (-4d-226)) .or. (.not. (t_1 <= 5d-302))) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + t_0
    else
        tmp = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0;
	double tmp;
	if ((t_1 <= -4e-226) || !(t_1 <= 5e-302)) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + t_0;
	} else {
		tmp = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0
	tmp = 0
	if (t_1 <= -4e-226) or not (t_1 <= 5e-302):
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + t_0
	else:
		tmp = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + t_0)
	tmp = 0.0
	if ((t_1 <= -4e-226) || !(t_1 <= 5e-302))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + t_0);
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT)));
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + t_0;
	tmp = 0.0;
	if ((t_1 <= -4e-226) || ~((t_1 <= 5e-302)))
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	else
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e-226], N[Not[LessEqual[t$95$1, 5e-302]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))))) < -3.99999999999999969e-226 or 5.00000000000000033e-302 < (+.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 EDonor around inf

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

      1. lower-/.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if -3.99999999999999969e-226 < (+.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))))) < 5.00000000000000033e-302

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 16.3%

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

   9. Taylor expanded in NdChar around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-+.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 95.3

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

   11. Applied rewrites 95.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -4 \cdot 10^{-226} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-302}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := t\_0 + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-185} \lor \neg \left(t\_1 \leq 10^{-238}\right):\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + \frac{\left(\left(\left(KbT + Vef\right) + Ev\right) + EAccept\right) - mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1
         (+
          t_0
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_1 -2e-185) (not (<= t_1 1e-238)))
     (+ t_0 (/ NaChar (+ 1.0 (/ (- (+ (+ (+ KbT Vef) Ev) EAccept) mu) KbT))))
     (/ NdChar (+ (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = t_0 + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_1 <= -2e-185) || !(t_1 <= 1e-238)) {
		tmp = t_0 + (NaChar / (1.0 + (((((KbT + Vef) + Ev) + EAccept) - mu) / KbT)));
	} else {
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_1 = t_0 + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_1 <= (-2d-185)) .or. (.not. (t_1 <= 1d-238))) then
        tmp = t_0 + (nachar / (1.0d0 + (((((kbt + vef) + ev) + eaccept) - mu) / kbt)))
    else
        tmp = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = t_0 + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_1 <= -2e-185) || !(t_1 <= 1e-238)) {
		tmp = t_0 + (NaChar / (1.0 + (((((KbT + Vef) + Ev) + EAccept) - mu) / KbT)));
	} else {
		tmp = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = t_0 + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_1 <= -2e-185) or not (t_1 <= 1e-238):
		tmp = t_0 + (NaChar / (1.0 + (((((KbT + Vef) + Ev) + EAccept) - mu) / KbT)))
	else:
		tmp = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_1 = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if ((t_1 <= -2e-185) || !(t_1 <= 1e-238))
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + Float64(Float64(Float64(Float64(Float64(KbT + Vef) + Ev) + EAccept) - mu) / KbT))));
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = t_0 + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_1 <= -2e-185) || ~((t_1 <= 1e-238)))
		tmp = t_0 + (NaChar / (1.0 + (((((KbT + Vef) + Ev) + EAccept) - mu) / KbT)));
	else
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-185], N[Not[LessEqual[t$95$1, 1e-238]], $MachinePrecision]], N[(t$95$0 + N[(NaChar / N[(1.0 + N[(N[(N[(N[(N[(KbT + Vef), $MachinePrecision] + Ev), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2e-185 or 9.9999999999999999e-239 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in KbT around 0

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

   8. Applied rewrites 68.0%

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

   if -2e-185 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999999e-239

   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

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 14.0%

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

   9. Taylor expanded in NdChar around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-+.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 86.1

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

   11. Applied rewrites 86.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-185} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 10^{-238}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \frac{\left(\left(\left(KbT + Vef\right) + Ev\right) + EAccept\right) - mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-62} \lor \neg \left(t\_1 \leq 200000000000\right):\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 t_0)))))
   (if (or (<= t_1 -2e-62) (not (<= t_1 200000000000.0)))
     (+ (* 0.5 NdChar) (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
     (/ NaChar (+ t_0 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	double tmp;
	if ((t_1 <= -2e-62) || !(t_1 <= 200000000000.0)) {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else {
		tmp = NaChar / (t_0 + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((((ev + vef) + eaccept) - mu) / kbt))
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + t_0))
    if ((t_1 <= (-2d-62)) .or. (.not. (t_1 <= 200000000000.0d0))) then
        tmp = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else
        tmp = nachar / (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	double tmp;
	if ((t_1 <= -2e-62) || !(t_1 <= 200000000000.0)) {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else {
		tmp = NaChar / (t_0 + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0))
	tmp = 0
	if (t_1 <= -2e-62) or not (t_1 <= 200000000000.0):
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	else:
		tmp = NaChar / (t_0 + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + t_0)))
	tmp = 0.0
	if ((t_1 <= -2e-62) || !(t_1 <= 200000000000.0))
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	else
		tmp = Float64(NaChar / Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((((Ev + Vef) + EAccept) - mu) / KbT));
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + t_0));
	tmp = 0.0;
	if ((t_1 <= -2e-62) || ~((t_1 <= 200000000000.0)))
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((EAccept / KbT))));
	else
		tmp = NaChar / (t_0 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-62], N[Not[LessEqual[t$95$1, 200000000000.0]], $MachinePrecision]], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.0000000000000001e-62 or 2e11 < (+.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

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

      1. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in Vef around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 63.3

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

   8. Applied rewrites 63.3%

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

   9. Taylor expanded in EAccept around inf

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

       1. lower-/.f64 53.5

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

   11. Applied rewrites 53.5%

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

   if -2.0000000000000001e-62 < (+.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))))) < 2e11

   1. Initial program 99.9%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 69.8

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

   5. Applied rewrites 69.8%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-62} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 200000000000\right):\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-62} \lor \neg \left(t\_0 \leq 200000000000\right):\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-62) (not (<= t_0 200000000000.0)))
     (+ (* 0.5 NdChar) (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
     (/ NaChar (+ (exp (/ (- (+ Vef Ev) mu) KbT)) 1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-62) || !(t_0 <= 200000000000.0)) {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else {
		tmp = NaChar / (exp((((Vef + Ev) - 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) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-2d-62)) .or. (.not. (t_0 <= 200000000000.0d0))) then
        tmp = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else
        tmp = nachar / (exp((((vef + ev) - 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 t_0 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-62) || !(t_0 <= 200000000000.0)) {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else {
		tmp = NaChar / (Math.exp((((Vef + Ev) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -2e-62) or not (t_0 <= 200000000000.0):
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	else:
		tmp = NaChar / (math.exp((((Vef + Ev) - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -2e-62) || !(t_0 <= 200000000000.0))
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -2e-62) || ~((t_0 <= 200000000000.0)))
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((EAccept / KbT))));
	else
		tmp = NaChar / (exp((((Vef + Ev) - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-62], N[Not[LessEqual[t$95$0, 200000000000.0]], $MachinePrecision]], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.0000000000000001e-62 or 2e11 < (+.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

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

      1. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in Vef around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 63.3

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

   8. Applied rewrites 63.3%

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

   9. Taylor expanded in EAccept around inf

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

       1. lower-/.f64 53.5

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

   11. Applied rewrites 53.5%

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

   if -2.0000000000000001e-62 < (+.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))))) < 2e11

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 99.9

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 69.8

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

   7. Applied rewrites 69.8%

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

   8. Taylor expanded in EAccept around 0

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

   10. Applied rewrites 61.6%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-62} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 200000000000\right):\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-215} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-300}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{NdChar - NaChar}{\left(-NaChar\right) \cdot NaChar}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-215) (not (<= t_0 2e-300)))
     (* 0.5 (+ NdChar NaChar))
     (* 0.5 (pow (/ (- NdChar NaChar) (* (- NaChar) NaChar)) -1.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-215) || !(t_0 <= 2e-300)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * pow(((NdChar - NaChar) / (-NaChar * NaChar)), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-2d-215)) .or. (.not. (t_0 <= 2d-300))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = 0.5d0 * (((ndchar - nachar) / (-nachar * nachar)) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-215) || !(t_0 <= 2e-300)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * Math.pow(((NdChar - NaChar) / (-NaChar * NaChar)), -1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -2e-215) or not (t_0 <= 2e-300):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = 0.5 * math.pow(((NdChar - NaChar) / (-NaChar * NaChar)), -1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -2e-215) || !(t_0 <= 2e-300))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(0.5 * (Float64(Float64(NdChar - NaChar) / Float64(Float64(-NaChar) * NaChar)) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -2e-215) || ~((t_0 <= 2e-300)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = 0.5 * (((NdChar - NaChar) / (-NaChar * NaChar)) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-215], N[Not[LessEqual[t$95$0, 2e-300]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[(N[(NdChar - NaChar), $MachinePrecision] / N[((-NaChar) * NaChar), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.00000000000000008e-215 or 2.00000000000000005e-300 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 36.7

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

   5. Applied rewrites 36.7%

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

   if -2.00000000000000008e-215 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.00000000000000005e-300

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   7. Applied rewrites 6.0%

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

   8. Taylor expanded in NdChar around 0

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

   10. Applied rewrites 24.6%

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

   12. Applied rewrites 24.6%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-215} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-300}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{NdChar - NaChar}{\left(-NaChar\right) \cdot NaChar}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-169} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-228}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-169) (not (<= t_0 2e-228)))
     (* 0.5 (+ NdChar NaChar))
     (/
      NaChar
      (- (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Ev KbT) (/ Vef KbT))) (/ mu KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-169) || !(t_0 <= 2e-228)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NaChar / (((2.0 + (EAccept / KbT)) + ((Ev / KbT) + (Vef / KbT))) - (mu / KbT));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-2d-169)) .or. (.not. (t_0 <= 2d-228))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = nachar / (((2.0d0 + (eaccept / kbt)) + ((ev / kbt) + (vef / kbt))) - (mu / kbt))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-169) || !(t_0 <= 2e-228)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = NaChar / (((2.0 + (EAccept / KbT)) + ((Ev / KbT) + (Vef / KbT))) - (mu / KbT));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -2e-169) or not (t_0 <= 2e-228):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = NaChar / (((2.0 + (EAccept / KbT)) + ((Ev / KbT) + (Vef / KbT))) - (mu / KbT))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -2e-169) || !(t_0 <= 2e-228))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(NaChar / Float64(Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Ev / KbT) + Float64(Vef / KbT))) - Float64(mu / KbT)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -2e-169) || ~((t_0 <= 2e-228)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = NaChar / (((2.0 + (EAccept / KbT)) + ((Ev / KbT) + (Vef / KbT))) - (mu / KbT));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-169], N[Not[LessEqual[t$95$0, 2e-228]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.00000000000000004e-169 or 2.00000000000000007e-228 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 39.0

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

   5. Applied rewrites 39.0%

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

   if -2.00000000000000004e-169 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.00000000000000007e-228

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 99.9

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 80.2

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

   7. Applied rewrites 80.2%

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

   8. Taylor expanded in KbT around inf

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

   10. Applied rewrites 34.8%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-169} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-228}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-237} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{\left(-NaChar\right) \cdot NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)} \cdot \left(NaChar + NdChar\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -1e-237) (not (<= t_0 0.0)))
     (* 0.5 (+ NdChar NaChar))
     (*
      0.5
      (*
       (/ (* (- NaChar) NaChar) (* (+ NaChar NdChar) (- NdChar NaChar)))
       (+ NaChar NdChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -1e-237) || !(t_0 <= 0.0)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * (((-NaChar * NaChar) / ((NaChar + NdChar) * (NdChar - NaChar))) * (NaChar + NdChar));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-1d-237)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = 0.5d0 * (((-nachar * nachar) / ((nachar + ndchar) * (ndchar - nachar))) * (nachar + ndchar))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -1e-237) || !(t_0 <= 0.0)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = 0.5 * (((-NaChar * NaChar) / ((NaChar + NdChar) * (NdChar - NaChar))) * (NaChar + NdChar));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -1e-237) or not (t_0 <= 0.0):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = 0.5 * (((-NaChar * NaChar) / ((NaChar + NdChar) * (NdChar - NaChar))) * (NaChar + NdChar))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -1e-237) || !(t_0 <= 0.0))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(Float64(-NaChar) * NaChar) / Float64(Float64(NaChar + NdChar) * Float64(NdChar - NaChar))) * Float64(NaChar + NdChar)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -1e-237) || ~((t_0 <= 0.0)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = 0.5 * (((-NaChar * NaChar) / ((NaChar + NdChar) * (NdChar - NaChar))) * (NaChar + NdChar));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-237], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[((-NaChar) * NaChar), $MachinePrecision] / N[(N[(NaChar + NdChar), $MachinePrecision] * N[(NdChar - NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(NaChar + NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999999e-238 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 35.6

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

   5. Applied rewrites 35.6%

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

   if -9.9999999999999999e-238 < (+.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 99.8%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

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

   7. Applied rewrites 6.1%

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

   8. Taylor expanded in NdChar around 0

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

   10. Applied rewrites 26.9%

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

   12. Applied rewrites 30.0%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -1 \cdot 10^{-237} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 0\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{\left(-NaChar\right) \cdot NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)} \cdot \left(NaChar + NdChar\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-215} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-300}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-NaChar\right) \cdot NaChar\right) \cdot 0.5}{NdChar - NaChar}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-215) (not (<= t_0 2e-300)))
     (* 0.5 (+ NdChar NaChar))
     (/ (* (* (- NaChar) NaChar) 0.5) (- NdChar NaChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-215) || !(t_0 <= 2e-300)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = ((-NaChar * NaChar) * 0.5) / (NdChar - NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-2d-215)) .or. (.not. (t_0 <= 2d-300))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = ((-nachar * nachar) * 0.5d0) / (ndchar - nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-215) || !(t_0 <= 2e-300)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = ((-NaChar * NaChar) * 0.5) / (NdChar - NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -2e-215) or not (t_0 <= 2e-300):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = ((-NaChar * NaChar) * 0.5) / (NdChar - NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -2e-215) || !(t_0 <= 2e-300))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(Float64(Float64(Float64(-NaChar) * NaChar) * 0.5) / Float64(NdChar - NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -2e-215) || ~((t_0 <= 2e-300)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = ((-NaChar * NaChar) * 0.5) / (NdChar - NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-215], N[Not[LessEqual[t$95$0, 2e-300]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-NaChar) * NaChar), $MachinePrecision] * 0.5), $MachinePrecision] / N[(NdChar - NaChar), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.00000000000000008e-215 or 2.00000000000000005e-300 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 36.7

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

   5. Applied rewrites 36.7%

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

   if -2.00000000000000008e-215 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.00000000000000005e-300

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   7. Applied rewrites 6.0%

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

   8. Taylor expanded in NdChar around 0

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

   10. Applied rewrites 24.6%

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

   12. Applied rewrites 24.6%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-215} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 2 \cdot 10^{-300}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-NaChar\right) \cdot NaChar\right) \cdot 0.5}{NdChar - NaChar}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-215} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-277}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(NaChar \cdot EAccept\right) \cdot -0.25}{KbT}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)))))))
   (if (or (<= t_0 -2e-215) (not (<= t_0 5e-277)))
     (* 0.5 (+ NdChar NaChar))
     (/ (* (* NaChar EAccept) -0.25) KbT))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-215) || !(t_0 <= 5e-277)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = ((NaChar * EAccept) * -0.25) / KbT;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) - mu) / kbt))))
    if ((t_0 <= (-2d-215)) .or. (.not. (t_0 <= 5d-277))) then
        tmp = 0.5d0 * (ndchar + nachar)
    else
        tmp = ((nachar * eaccept) * (-0.25d0)) / kbt
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	double tmp;
	if ((t_0 <= -2e-215) || !(t_0 <= 5e-277)) {
		tmp = 0.5 * (NdChar + NaChar);
	} else {
		tmp = ((NaChar * EAccept) * -0.25) / KbT;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) - mu) / KbT))))
	tmp = 0
	if (t_0 <= -2e-215) or not (t_0 <= 5e-277):
		tmp = 0.5 * (NdChar + NaChar)
	else:
		tmp = ((NaChar * EAccept) * -0.25) / KbT
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if ((t_0 <= -2e-215) || !(t_0 <= 5e-277))
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	else
		tmp = Float64(Float64(Float64(NaChar * EAccept) * -0.25) / KbT);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if ((t_0 <= -2e-215) || ~((t_0 <= 5e-277)))
		tmp = 0.5 * (NdChar + NaChar);
	else
		tmp = ((NaChar * EAccept) * -0.25) / KbT;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-215], N[Not[LessEqual[t$95$0, 5e-277]], $MachinePrecision]], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], N[(N[(N[(NaChar * EAccept), $MachinePrecision] * -0.25), $MachinePrecision] / KbT), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.00000000000000008e-215 or 5e-277 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 37.3

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

   5. Applied rewrites 37.3%

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

   if -2.00000000000000008e-215 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5e-277

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 15.5%

      \[\leadsto \left(EAccept \cdot \frac{NaChar}{KbT}\right) \cdot \color{blue}{-0.25} \]
   9. Step-by-step derivation

   10. Applied rewrites 17.4%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq -2 \cdot 10^{-215} \lor \neg \left(\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}}} \leq 5 \cdot 10^{-277}\right):\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(NaChar \cdot EAccept\right) \cdot -0.25}{KbT}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{3}\right)}^{\left(0.3333333333333333 \cdot \left(\left(2 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right) \cdot \frac{0.5}{KbT}\right)\right)}} \end{array} \]

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
    (pow
     (exp 3.0)
     (*
      0.3333333333333333
      (* (* 2.0 (- (+ EAccept (+ Ev Vef)) mu)) (/ 0.5 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 + pow(exp(3.0), (0.3333333333333333 * ((2.0 * ((EAccept + (Ev + Vef)) - mu)) * (0.5 / 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(3.0d0) ** (0.3333333333333333d0 * ((2.0d0 * ((eaccept + (ev + vef)) - mu)) * (0.5d0 / 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.pow(Math.exp(3.0), (0.3333333333333333 * ((2.0 * ((EAccept + (Ev + Vef)) - mu)) * (0.5 / 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.pow(math.exp(3.0), (0.3333333333333333 * ((2.0 * ((EAccept + (Ev + Vef)) - mu)) * (0.5 / 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(3.0) ^ Float64(0.3333333333333333 * Float64(Float64(2.0 * Float64(Float64(EAccept + Float64(Ev + Vef)) - mu)) * Float64(0.5 / 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(3.0) ^ (0.3333333333333333 * ((2.0 * ((EAccept + (Ev + Vef)) - mu)) * (0.5 / 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[Power[N[Exp[3.0], $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 * N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] * N[(0.5 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 100.0

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

   10. lift-+.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 100.0

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. sub-neg N/A

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

   17. lift-neg.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. associate-/l/ N/A

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

   21. lower-/.f64 N/A

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{{\left(e^{3}\right)}^{\left(0.3333333333333333 \cdot \left(\left(2 \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right) \cdot \frac{0.5}{KbT}\right)\right)}}} \]
9. Add Preprocessing

Alternative 14: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + {\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{2 \cdot KbT}\right)}} \end{array} \]

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 (pow (* (E) (E)) (/ (- (+ (+ Ev Vef) EAccept) mu) (* 2.0 KbT)))))))

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 100.0

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

   10. lift-+.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 100.0

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. exp-1-e N/A

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

   8. lower-E.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. sub-neg N/A

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

   17. lift-neg.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. associate-/l/ N/A

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

   21. lower-/.f64 N/A

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

6. Applied rewrites 100.0%

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

Alternative 15: 100.0% accurate, 1.0× speedup

Math

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

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 (pow (E) (/ (- (+ (+ Ev Vef) EAccept) mu) KbT))))))

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 100.0

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

   10. lift-+.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 100.0

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f64 100.0

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

   4. lift-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. div-sub N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. div-sub N/A

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

   11. lift--.f64 N/A

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

   12. lift-/.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 16: 67.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -7.8 \cdot 10^{-113} \lor \neg \left(NdChar \leq 3.7 \cdot 10^{-112}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -7.8e-113) (not (<= NdChar 3.7e-112)))
   (/ NdChar (+ (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)) 1.0))
   (/ NaChar (+ (exp (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -7.8e-113) || !(NdChar <= 3.7e-112)) {
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-7.8d-113)) .or. (.not. (ndchar <= 3.7d-112))) then
        tmp = ndchar / (exp(((((mu + vef) + edonor) - ec) / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp(((((ev + vef) + eaccept) - mu) / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -7.8e-113) || !(NdChar <= 3.7e-112)) {
		tmp = NdChar / (Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -7.8e-113) or not (NdChar <= 3.7e-112):
		tmp = NdChar / (math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -7.8e-113) || !(NdChar <= 3.7e-112))
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -7.8e-113) || ~((NdChar <= 3.7e-112)))
		tmp = NdChar / (exp(((((mu + Vef) + EDonor) - Ec) / KbT)) + 1.0);
	else
		tmp = NaChar / (exp(((((Ev + Vef) + EAccept) - mu) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -7.8e-113], N[Not[LessEqual[NdChar, 3.7e-112]], $MachinePrecision]], N[(NdChar / N[(N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -7.7999999999999997e-113 or 3.6999999999999998e-112 < NdChar

   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

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 22.2%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 5.2%

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

   9. Taylor expanded in NdChar around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-exp.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-+.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 72.4

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

   11. Applied rewrites 72.4%

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

   if -7.7999999999999997e-113 < NdChar < 3.6999999999999998e-112

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -7.8 \cdot 10^{-113} \lor \neg \left(NdChar \leq 3.7 \cdot 10^{-112}\right):\\ \;\;\;\;\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -8.5 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right), t\_0\right)\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, EAccept \cdot \frac{NaChar}{KbT}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -8.5e+210)
     (fma
      -0.25
      (fma
       NaChar
       (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)
       (* NdChar (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)))
      t_0)
     (if (<= KbT 2.6e+157)
       (/ NaChar (+ (exp (/ (- (+ Vef Ev) mu) KbT)) 1.0))
       (fma -0.25 (* EAccept (/ NaChar 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 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -8.5e+210) {
		tmp = fma(-0.25, fma(NaChar, ((((Ev + Vef) + EAccept) - mu) / KbT), (NdChar * ((((mu + Vef) + EDonor) - Ec) / KbT))), t_0);
	} else if (KbT <= 2.6e+157) {
		tmp = NaChar / (exp((((Vef + Ev) - mu) / KbT)) + 1.0);
	} else {
		tmp = fma(-0.25, (EAccept * (NaChar / KbT)), t_0);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -8.5e+210)
		tmp = fma(-0.25, fma(NaChar, Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT), Float64(NdChar * Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT))), t_0);
	elseif (KbT <= 2.6e+157)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)) + 1.0));
	else
		tmp = fma(-0.25, Float64(EAccept * Float64(NaChar / KbT)), 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[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -8.5e+210], N[(-0.25 * N[(NaChar * N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision] + N[(NdChar * N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[KbT, 2.6e+157], N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(EAccept * N[(NaChar / KbT), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -8.49999999999999975e210

   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

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 86.7%

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

   if -8.49999999999999975e210 < KbT < 2.60000000000000011e157

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 100.0

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 59.7

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

   7. Applied rewrites 59.7%

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

   8. Taylor expanded in EAccept around 0

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

   10. Applied rewrites 51.8%

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

   if 2.60000000000000011e157 < 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

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 71.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -8.5 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right), 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + Ev\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, EAccept \cdot \frac{NaChar}{KbT}, 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -9.6 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right), 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \mathbf{elif}\;KbT \leq 6.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -9.6e+158)
   (fma
    -0.25
    (fma
     NaChar
     (/ (- (+ (+ Ev Vef) EAccept) mu) KbT)
     (* NdChar (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)))
    (* 0.5 (+ NdChar NaChar)))
   (if (<= KbT 6.6e+77)
     (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
     (+
      (* 0.5 NdChar)
      (/
       NaChar
       (-
        (+ (+ 2.0 (/ EAccept KbT)) (+ (/ Ev KbT) (/ Vef KbT)))
        (/ mu KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -9.6e+158) {
		tmp = fma(-0.25, fma(NaChar, ((((Ev + Vef) + EAccept) - mu) / KbT), (NdChar * ((((mu + Vef) + EDonor) - Ec) / KbT))), (0.5 * (NdChar + NaChar)));
	} else if (KbT <= 6.6e+77) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (((2.0 + (EAccept / KbT)) + ((Ev / KbT) + (Vef / KbT))) - (mu / KbT)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -9.6e+158)
		tmp = fma(-0.25, fma(NaChar, Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) - mu) / KbT), Float64(NdChar * Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT))), Float64(0.5 * Float64(NdChar + NaChar)));
	elseif (KbT <= 6.6e+77)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(Float64(Float64(2.0 + Float64(EAccept / KbT)) + Float64(Float64(Ev / KbT) + Float64(Vef / KbT))) - Float64(mu / KbT))));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -9.6e+158], N[(-0.25 * N[(NaChar * N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision] + N[(NdChar * N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 6.6e+77], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(N[(N[(2.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -9.60000000000000033e158

   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

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   if -9.60000000000000033e158 < KbT < 6.5999999999999996e77

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 99.9

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 59.8

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

   7. Applied rewrites 59.8%

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

   8. Taylor expanded in EAccept around inf

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

   10. Applied rewrites 31.6%

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

   if 6.5999999999999996e77 < 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

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

      1. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 56.4

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

   8. Applied rewrites 56.4%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -9.6 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right), 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \mathbf{elif}\;KbT \leq 6.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept 5e-161)
   (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
   (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 5e-161) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (eaccept <= 5d-161) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 5e-161) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EAccept <= 5e-161:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	else:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EAccept <= 5e-161)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	else
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EAccept <= 5e-161)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	else
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EAccept, 5e-161], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if EAccept < 4.9999999999999999e-161

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 99.9

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 59.0

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

   7. Applied rewrites 59.0%

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

   8. Taylor expanded in Ev around inf

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

   10. Applied rewrites 42.8%

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

   if 4.9999999999999999e-161 < EAccept

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 100.0

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 54.2

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

   7. Applied rewrites 54.2%

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

   8. Taylor expanded in EAccept around inf

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

   10. Applied rewrites 41.8%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 22.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.5 \cdot 10^{-96} \lor \neg \left(NdChar \leq 1.6 \cdot 10^{-82}\right):\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -2.5e-96) (not (<= NdChar 1.6e-82)))
   (* 0.5 NdChar)
   (* 0.5 NaChar)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.5e-96) || !(NdChar <= 1.6e-82)) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = 0.5 * NaChar;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-2.5d-96)) .or. (.not. (ndchar <= 1.6d-82))) then
        tmp = 0.5d0 * ndchar
    else
        tmp = 0.5d0 * nachar
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.5e-96) || !(NdChar <= 1.6e-82)) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = 0.5 * NaChar;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -2.5e-96) or not (NdChar <= 1.6e-82):
		tmp = 0.5 * NdChar
	else:
		tmp = 0.5 * NaChar
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -2.5e-96) || !(NdChar <= 1.6e-82))
		tmp = Float64(0.5 * NdChar);
	else
		tmp = Float64(0.5 * NaChar);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -2.5e-96) || ~((NdChar <= 1.6e-82)))
		tmp = 0.5 * NdChar;
	else
		tmp = 0.5 * NaChar;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -2.5e-96], N[Not[LessEqual[NdChar, 1.6e-82]], $MachinePrecision]], N[(0.5 * NdChar), $MachinePrecision], N[(0.5 * NaChar), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.49999999999999997e-96 or 1.6000000000000001e-82 < NdChar

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 28.9

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

   5. Applied rewrites 28.9%

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

   7. Applied rewrites 17.8%

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

   8. Taylor expanded in NdChar around 0

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

   10. Applied rewrites 11.0%

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

   11. Taylor expanded in NdChar around inf

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

   13. Applied rewrites 24.8%

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

   if -2.49999999999999997e-96 < NdChar < 1.6000000000000001e-82

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 28.6

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in NdChar around 0

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

   8. Applied rewrites 29.8%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.5 \cdot 10^{-96} \lor \neg \left(NdChar \leq 1.6 \cdot 10^{-82}\right):\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NaChar\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 27.9% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 28.8

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

5. Applied rewrites 28.8%

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

Alternative 22: 18.6% accurate, 46.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in KbT around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 28.8

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

5. Applied rewrites 28.8%

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

7. Applied rewrites 19.4%

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

8. Taylor expanded in NdChar around 0

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

10. Applied rewrites 15.1%

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

11. Taylor expanded in NdChar around inf

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

13. Applied rewrites 20.0%

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

14. Final simplification 20.0%

    \[\leadsto 0.5 \cdot NdChar \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024299 
(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))))))