Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 16.3s

Alternatives: 20

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 3.7204937099110756e+40
  2. Ec = -3.6999643915367085e+284
  3. Vef = 3.166821933474963e-302
  4. EDonor = 1.0313083870286259e-17
  5. mu = 2.733773590999487e+118
  6. KbT = -7.399737273829446e+146
  7. NaChar = -1.0110068820921685e+41
  8. Ev = 1.7579796419385627e+177
  9. EAccept = -1.435601312384359e+102

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}\\ t_1 := \left(mu + Vef\right) + EDonor\\ t_2 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_3 := \frac{NaChar}{-1 - t\_2}\\ t_4 := \frac{NdChar}{1 + t\_0} - t\_3\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\frac{NaChar}{1 + 1} - \frac{NdChar}{-1 - t\_0}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-120}:\\ \;\;\;\;\left(\frac{1}{1 + t\_2} - \frac{NdChar}{\left(-1 - \left(1 - \frac{Ec - t\_1}{KbT}\right)\right) \cdot NaChar}\right) \cdot NaChar\\ \mathbf{elif}\;t\_4 \leq 10^{-72}:\\ \;\;\;\;\frac{NdChar}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{t\_1 - Ec}{KbT}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar - t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)))
        (t_1 (+ (+ mu Vef) EDonor))
        (t_2 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_3 (/ NaChar (- -1.0 t_2)))
        (t_4 (- (/ NdChar (+ 1.0 t_0)) t_3)))
   (if (<= t_4 -2e+48)
     (- (/ NaChar (+ 1.0 1.0)) (/ NdChar (- -1.0 t_0)))
     (if (<= t_4 -5e-120)
       (*
        (-
         (/ 1.0 (+ 1.0 t_2))
         (/ NdChar (* (- -1.0 (- 1.0 (/ (- Ec t_1) KbT))) NaChar)))
        NaChar)
       (if (<= t_4 1e-72)
         (/ NdChar (+ 1.0 (pow (E) (/ (- t_1 Ec) KbT))))
         (- (* 0.5 NdChar) t_3))))))

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))))) < -2.00000000000000009e48

   1. Initial program 99.7%

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

   3. Taylor expanded in KbT around inf

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

   5. Applied rewrites 72.7%

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

   if -2.00000000000000009e48 < (+.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))))) < -5.00000000000000007e-120

   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

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

   5. Applied rewrites 57.0%

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

      1. lift-exp.f64 N/A

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

      2. *-lft-identity N/A

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

      3. pow-exp N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-pow.f64 57.0

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

      6. lift-exp.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in NaChar around inf

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

   10. Applied rewrites 96.9%

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

   11. Taylor expanded in KbT around inf

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

   13. Applied rewrites 75.0%

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

   if -5.00000000000000007e-120 < (+.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))))) < 9.9999999999999997e-73

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

   5. Applied rewrites 14.8%

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

      1. lift-exp.f64 N/A

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

      2. *-lft-identity N/A

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

      3. pow-exp N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-pow.f64 14.8

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

      6. lift-exp.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 14.8

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

   7. Applied rewrites 14.8%

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

   8. Taylor expanded in NaChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   10. Applied rewrites 82.0%

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

   if 9.9999999999999997e-73 < (+.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 74.4

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

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\frac{NaChar}{1 + 1} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -5 \cdot 10^{-120}:\\ \;\;\;\;\left(\frac{1}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} - \frac{NdChar}{\left(-1 - \left(1 - \frac{Ec - \left(\left(mu + Vef\right) + EDonor\right)}{KbT}\right)\right) \cdot NaChar}\right) \cdot NaChar\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 10^{-72}:\\ \;\;\;\;\frac{NdChar}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}\\ t_1 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - t\_1}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-292}:\\ \;\;\;\;\frac{NaChar}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NdChar (+ 1.0 (exp (/ (- EDonor Ec) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept Ev) mu) KbT))))))
        (t_1 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_2
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 t_1)))))
   (if (<= t_2 -4e-308) t_0 (if (<= t_2 2e-292) (/ NaChar (+ 1.0 t_1)) 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 = (NdChar / (1.0 + exp(((EDonor - Ec) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + Ev) - mu) / KbT))));
	double t_1 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_2 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - t_1));
	double tmp;
	if (t_2 <= -4e-308) {
		tmp = t_0;
	} else if (t_2 <= 2e-292) {
		tmp = NaChar / (1.0 + t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp(((edonor - ec) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + ev) - mu) / kbt))))
    t_1 = exp((((eaccept + (ev + vef)) - mu) / kbt))
    t_2 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - t_1))
    if (t_2 <= (-4d-308)) then
        tmp = t_0
    else if (t_2 <= 2d-292) then
        tmp = nachar / (1.0d0 + t_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp(((EDonor - Ec) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + Ev) - mu) / KbT))));
	double t_1 = Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_2 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - t_1));
	double tmp;
	if (t_2 <= -4e-308) {
		tmp = t_0;
	} else if (t_2 <= 2e-292) {
		tmp = NaChar / (1.0 + t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp(((EDonor - Ec) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + Ev) - mu) / KbT))))
	t_1 = math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))
	t_2 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - t_1))
	tmp = 0
	if t_2 <= -4e-308:
		tmp = t_0
	elif t_2 <= 2e-292:
		tmp = NaChar / (1.0 + t_1)
	else:
		tmp = t_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(EDonor - Ec) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)))))
	t_1 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - t_1)))
	tmp = 0.0
	if (t_2 <= -4e-308)
		tmp = t_0;
	elseif (t_2 <= 2e-292)
		tmp = Float64(NaChar / Float64(1.0 + t_1));
	else
		tmp = t_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(((EDonor - Ec) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + Ev) - mu) / KbT))));
	t_1 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	t_2 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - t_1));
	tmp = 0.0;
	if (t_2 <= -4e-308)
		tmp = t_0;
	elseif (t_2 <= 2e-292)
		tmp = NaChar / (1.0 + t_1);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-308], t$95$0, If[LessEqual[t$95$2, 2e-292], N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.00000000000000013e-308 or 2.0000000000000001e-292 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in mu around 0

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

   8. Applied rewrites 81.5%

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

   if -4.00000000000000013e-308 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 2.0000000000000001e-292

   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 NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -4 \cdot 10^{-308}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 2 \cdot 10^{-292}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))
          (/ NdChar (- -1.0 (exp (/ (- (+ mu EDonor) Ec) KbT))))))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -2e-259)
     t_0
     (if (<= t_1 0.0)
       (/ NdChar (+ 1.0 (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + exp((-mu / KbT)))) - (NdChar / (-1.0 - exp((((mu + EDonor) - Ec) / KbT))));
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar / (1.0d0 + exp((-mu / kbt)))) - (ndchar / ((-1.0d0) - exp((((mu + edonor) - ec) / kbt))))
    t_1 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    if (t_1 <= (-2d-259)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = ndchar / (1.0d0 + exp(((((mu + vef) + edonor) - ec) / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + Math.exp((-mu / KbT)))) - (NdChar / (-1.0 - Math.exp((((mu + EDonor) - Ec) / KbT))));
	double t_1 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = NdChar / (1.0 + Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (1.0 + math.exp((-mu / KbT)))) - (NdChar / (-1.0 - math.exp((((mu + EDonor) - Ec) / KbT))))
	t_1 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	tmp = 0
	if t_1 <= -2e-259:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = NdChar / (1.0 + math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(Float64(mu + EDonor) - Ec) / KbT)))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (1.0 + exp((-mu / KbT)))) - (NdChar / (-1.0 - exp((((mu + EDonor) - Ec) / KbT))));
	t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -2e-259)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-259], t$95$0, If[LessEqual[t$95$1, 0.0], N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.0000000000000001e-259 or 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in mu around inf

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

   8. Applied rewrites 76.4%

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

   if -2.0000000000000001e-259 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in NaChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-148}:\\ \;\;\;\;\frac{NdChar}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
          (/ NdChar (- -1.0 (exp (/ (- (+ mu EDonor) Ec) KbT))))))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -5e-120)
     t_0
     (if (<= t_1 1e-148)
       (/ NdChar (+ 1.0 (pow (E) (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))))
       t_0))))

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000007e-120 or 9.99999999999999936e-149 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in Ev around inf

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

   8. Applied rewrites 71.8%

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

   if -5.00000000000000007e-120 < (+.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))))) < 9.99999999999999936e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

   5. Applied rewrites 11.2%

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

      1. lift-exp.f64 N/A

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

      2. *-lft-identity N/A

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

      3. pow-exp N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-pow.f64 11.2

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

      6. lift-exp.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 11.2

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

   7. Applied rewrites 11.2%

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

   8. Taylor expanded in NaChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   10. Applied rewrites 84.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -5 \cdot 10^{-120}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 10^{-148}:\\ \;\;\;\;\frac{NdChar}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-243}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\frac{NdChar}{2 - \left(\frac{Ec - mu}{KbT} - \frac{EDonor}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -5e-243)
     t_0
     (if (<= t_1 5e-157)
       (/ NdChar (- 2.0 (- (/ (- Ec mu) KbT) (/ EDonor KbT))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -5e-243) {
		tmp = t_0;
	} else if (t_1 <= 5e-157) {
		tmp = NdChar / (2.0 - (((Ec - mu) / KbT) - (EDonor / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    if (t_1 <= (-5d-243)) then
        tmp = t_0
    else if (t_1 <= 5d-157) then
        tmp = ndchar / (2.0d0 - (((ec - mu) / kbt) - (edonor / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -5e-243) {
		tmp = t_0;
	} else if (t_1 <= 5e-157) {
		tmp = NdChar / (2.0 - (((Ec - mu) / KbT) - (EDonor / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	tmp = 0
	if t_1 <= -5e-243:
		tmp = t_0
	elif t_1 <= 5e-157:
		tmp = NdChar / (2.0 - (((Ec - mu) / KbT) - (EDonor / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -5e-243)
		tmp = t_0;
	elseif (t_1 <= 5e-157)
		tmp = Float64(NdChar / Float64(2.0 - Float64(Float64(Float64(Ec - mu) / KbT) - Float64(EDonor / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -5e-243)
		tmp = t_0;
	elseif (t_1 <= 5e-157)
		tmp = NdChar / (2.0 - (((Ec - mu) / KbT) - (EDonor / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-243], t$95$0, If[LessEqual[t$95$1, 5e-157], N[(NdChar / N[(2.0 - N[(N[(N[(Ec - mu), $MachinePrecision] / KbT), $MachinePrecision] - N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5e-243 or 5.0000000000000002e-157 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 37.5

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

   5. Applied rewrites 37.5%

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

   if -5e-243 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5.0000000000000002e-157

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 70.2%

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

   9. Taylor expanded in KbT around inf

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

   11. Applied rewrites 39.4%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -5 \cdot 10^{-243}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\frac{NdChar}{2 - \left(\frac{Ec - mu}{KbT} - \frac{EDonor}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + pow(exp(-1.0), ((((Ec - Vef) - EDonor) - mu) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - 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((-1.0d0)) ** ((((ec - vef) - edonor) - mu) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. distribute-frac-neg N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 8: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{if}\;Vef \leq -9.5 \cdot 10^{+211}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 5.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NaChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NdChar (- -1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)))))))
   (if (<= Vef -9.5e+211)
     t_0
     (if (<= Vef 5.8e+144)
       (-
        (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept Ev) mu) KbT))))
        (/ NdChar (- -1.0 (exp (/ (- (+ mu EDonor) Ec) KbT)))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + exp((Vef / KbT)))) - (NdChar / (-1.0 - exp(((mu - ((Ec - Vef) - EDonor)) / KbT))));
	double tmp;
	if (Vef <= -9.5e+211) {
		tmp = t_0;
	} else if (Vef <= 5.8e+144) {
		tmp = (NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT)))) - (NdChar / (-1.0 - exp((((mu + EDonor) - Ec) / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (nachar / (1.0d0 + exp((vef / kbt)))) - (ndchar / ((-1.0d0) - exp(((mu - ((ec - vef) - edonor)) / kbt))))
    if (vef <= (-9.5d+211)) then
        tmp = t_0
    else if (vef <= 5.8d+144) then
        tmp = (nachar / (1.0d0 + exp((((eaccept + ev) - mu) / kbt)))) - (ndchar / ((-1.0d0) - exp((((mu + edonor) - ec) / kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + Math.exp((Vef / KbT)))) - (NdChar / (-1.0 - Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))));
	double tmp;
	if (Vef <= -9.5e+211) {
		tmp = t_0;
	} else if (Vef <= 5.8e+144) {
		tmp = (NaChar / (1.0 + Math.exp((((EAccept + Ev) - mu) / KbT)))) - (NdChar / (-1.0 - Math.exp((((mu + EDonor) - Ec) / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (1.0 + math.exp((Vef / KbT)))) - (NdChar / (-1.0 - math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))))
	tmp = 0
	if Vef <= -9.5e+211:
		tmp = t_0
	elif Vef <= 5.8e+144:
		tmp = (NaChar / (1.0 + math.exp((((EAccept + Ev) - mu) / KbT)))) - (NdChar / (-1.0 - math.exp((((mu + EDonor) - Ec) / KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))))
	tmp = 0.0
	if (Vef <= -9.5e+211)
		tmp = t_0;
	elseif (Vef <= 5.8e+144)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(Float64(mu + EDonor) - Ec) / KbT)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (1.0 + exp((Vef / KbT)))) - (NdChar / (-1.0 - exp(((mu - ((Ec - Vef) - EDonor)) / KbT))));
	tmp = 0.0;
	if (Vef <= -9.5e+211)
		tmp = t_0;
	elseif (Vef <= 5.8e+144)
		tmp = (NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT)))) - (NdChar / (-1.0 - exp((((mu + EDonor) - Ec) / KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -9.5e+211], t$95$0, If[LessEqual[Vef, 5.8e+144], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -9.4999999999999997e211 or 5.79999999999999996e144 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around inf

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

      1. lower-/.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if -9.4999999999999997e211 < Vef < 5.79999999999999996e144

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 95.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -9.5 \cdot 10^{+211}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 5.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{if}\;Vef \leq -2.7 \cdot 10^{+169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 4.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NaChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NdChar (- -1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)))))))
   (if (<= Vef -2.7e+169)
     t_0
     (if (<= Vef 4.1e+74)
       (-
        (/ NdChar (+ 1.0 (exp (/ (- EDonor Ec) KbT))))
        (/ NaChar (- -1.0 (exp (/ (- (+ EAccept Ev) mu) 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((Vef / KbT)))) - (NdChar / (-1.0 - exp(((mu - ((Ec - Vef) - EDonor)) / KbT))));
	double tmp;
	if (Vef <= -2.7e+169) {
		tmp = t_0;
	} else if (Vef <= 4.1e+74) {
		tmp = (NdChar / (1.0 + exp(((EDonor - Ec) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + Ev) - mu) / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (nachar / (1.0d0 + exp((vef / kbt)))) - (ndchar / ((-1.0d0) - exp(((mu - ((ec - vef) - edonor)) / kbt))))
    if (vef <= (-2.7d+169)) then
        tmp = t_0
    else if (vef <= 4.1d+74) then
        tmp = (ndchar / (1.0d0 + exp(((edonor - ec) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + ev) - mu) / kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + Math.exp((Vef / KbT)))) - (NdChar / (-1.0 - Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))));
	double tmp;
	if (Vef <= -2.7e+169) {
		tmp = t_0;
	} else if (Vef <= 4.1e+74) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor - Ec) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + Ev) - mu) / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (1.0 + math.exp((Vef / KbT)))) - (NdChar / (-1.0 - math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))))
	tmp = 0
	if Vef <= -2.7e+169:
		tmp = t_0
	elif Vef <= 4.1e+74:
		tmp = (NdChar / (1.0 + math.exp(((EDonor - Ec) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + Ev) - mu) / KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))))
	tmp = 0.0
	if (Vef <= -2.7e+169)
		tmp = t_0;
	elseif (Vef <= 4.1e+74)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Ec) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (1.0 + exp((Vef / KbT)))) - (NdChar / (-1.0 - exp(((mu - ((Ec - Vef) - EDonor)) / KbT))));
	tmp = 0.0;
	if (Vef <= -2.7e+169)
		tmp = t_0;
	elseif (Vef <= 4.1e+74)
		tmp = (NdChar / (1.0 + exp(((EDonor - Ec) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + Ev) - mu) / KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -2.7e+169], t$95$0, If[LessEqual[Vef, 4.1e+74], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -2.69999999999999991e169 or 4.1e74 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around inf

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

      1. lower-/.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -2.69999999999999991e169 < Vef < 4.1e74

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in mu around 0

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

   8. Applied rewrites 88.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -2.7 \cdot 10^{+169}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 4.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.9%

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

Alternative 11: 44.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ t_2 := \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Vef \leq -4.6 \cdot 10^{+96}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Vef \leq -6.7 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 1.8 \cdot 10^{-113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq 1.95 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (- mu) KbT)))))
        (t_2 (/ NaChar (+ 1.0 (exp (/ Vef KbT))))))
   (if (<= Vef -4.6e+96)
     t_2
     (if (<= Vef -6.7e-16)
       t_1
       (if (<= Vef 2.1e-158)
         t_0
         (if (<= Vef 1.8e-113) t_1 (if (<= Vef 1.95e+46) t_0 t_2)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((EDonor / KbT)));
	double t_1 = NaChar / (1.0 + exp((-mu / KbT)));
	double t_2 = NaChar / (1.0 + exp((Vef / KbT)));
	double tmp;
	if (Vef <= -4.6e+96) {
		tmp = t_2;
	} else if (Vef <= -6.7e-16) {
		tmp = t_1;
	} else if (Vef <= 2.1e-158) {
		tmp = t_0;
	} else if (Vef <= 1.8e-113) {
		tmp = t_1;
	} else if (Vef <= 1.95e+46) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((edonor / kbt)))
    t_1 = nachar / (1.0d0 + exp((-mu / kbt)))
    t_2 = nachar / (1.0d0 + exp((vef / kbt)))
    if (vef <= (-4.6d+96)) then
        tmp = t_2
    else if (vef <= (-6.7d-16)) then
        tmp = t_1
    else if (vef <= 2.1d-158) then
        tmp = t_0
    else if (vef <= 1.8d-113) then
        tmp = t_1
    else if (vef <= 1.95d+46) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp((-mu / KbT)));
	double t_2 = NaChar / (1.0 + Math.exp((Vef / KbT)));
	double tmp;
	if (Vef <= -4.6e+96) {
		tmp = t_2;
	} else if (Vef <= -6.7e-16) {
		tmp = t_1;
	} else if (Vef <= 2.1e-158) {
		tmp = t_0;
	} else if (Vef <= 1.8e-113) {
		tmp = t_1;
	} else if (Vef <= 1.95e+46) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((EDonor / KbT)))
	t_1 = NaChar / (1.0 + math.exp((-mu / KbT)))
	t_2 = NaChar / (1.0 + math.exp((Vef / KbT)))
	tmp = 0
	if Vef <= -4.6e+96:
		tmp = t_2
	elif Vef <= -6.7e-16:
		tmp = t_1
	elif Vef <= 2.1e-158:
		tmp = t_0
	elif Vef <= 1.8e-113:
		tmp = t_1
	elif Vef <= 1.95e+46:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT))))
	t_2 = Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT))))
	tmp = 0.0
	if (Vef <= -4.6e+96)
		tmp = t_2;
	elseif (Vef <= -6.7e-16)
		tmp = t_1;
	elseif (Vef <= 2.1e-158)
		tmp = t_0;
	elseif (Vef <= 1.8e-113)
		tmp = t_1;
	elseif (Vef <= 1.95e+46)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((EDonor / KbT)));
	t_1 = NaChar / (1.0 + exp((-mu / KbT)));
	t_2 = NaChar / (1.0 + exp((Vef / KbT)));
	tmp = 0.0;
	if (Vef <= -4.6e+96)
		tmp = t_2;
	elseif (Vef <= -6.7e-16)
		tmp = t_1;
	elseif (Vef <= 2.1e-158)
		tmp = t_0;
	elseif (Vef <= 1.8e-113)
		tmp = t_1;
	elseif (Vef <= 1.95e+46)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -4.6e+96], t$95$2, If[LessEqual[Vef, -6.7e-16], t$95$1, If[LessEqual[Vef, 2.1e-158], t$95$0, If[LessEqual[Vef, 1.8e-113], t$95$1, If[LessEqual[Vef, 1.95e+46], t$95$0, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -4.6000000000000003e96 or 1.94999999999999997e46 < Vef

   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 NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 59.8%

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

   if -4.6000000000000003e96 < Vef < -6.7000000000000004e-16 or 2.09999999999999991e-158 < Vef < 1.79999999999999987e-113

   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 NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in mu around inf

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

   8. Applied rewrites 62.2%

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

   if -6.7000000000000004e-16 < Vef < 2.09999999999999991e-158 or 1.79999999999999987e-113 < Vef < 1.94999999999999997e46

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 68.8%

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

   9. Taylor expanded in EDonor around inf

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

   11. Applied rewrites 51.4%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -4.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq -6.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.95 \cdot 10^{+46}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Vef \leq -9.8 \cdot 10^{+213}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 2.35 \cdot 10^{-114}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 6 \cdot 10^{+191}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (+ mu EDonor) Ec) KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ Vef KbT))))))
   (if (<= Vef -9.8e+213)
     t_1
     (if (<= Vef 2.1e-158)
       t_0
       (if (<= Vef 2.35e-114)
         (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))
         (if (<= Vef 6e+191) t_0 t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((((mu + EDonor) - Ec) / KbT)));
	double t_1 = NaChar / (1.0 + exp((Vef / KbT)));
	double tmp;
	if (Vef <= -9.8e+213) {
		tmp = t_1;
	} else if (Vef <= 2.1e-158) {
		tmp = t_0;
	} else if (Vef <= 2.35e-114) {
		tmp = NaChar / (1.0 + exp((-mu / KbT)));
	} else if (Vef <= 6e+191) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((((mu + edonor) - ec) / kbt)))
    t_1 = nachar / (1.0d0 + exp((vef / kbt)))
    if (vef <= (-9.8d+213)) then
        tmp = t_1
    else if (vef <= 2.1d-158) then
        tmp = t_0
    else if (vef <= 2.35d-114) then
        tmp = nachar / (1.0d0 + exp((-mu / kbt)))
    else if (vef <= 6d+191) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((((mu + EDonor) - Ec) / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp((Vef / KbT)));
	double tmp;
	if (Vef <= -9.8e+213) {
		tmp = t_1;
	} else if (Vef <= 2.1e-158) {
		tmp = t_0;
	} else if (Vef <= 2.35e-114) {
		tmp = NaChar / (1.0 + Math.exp((-mu / KbT)));
	} else if (Vef <= 6e+191) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((((mu + EDonor) - Ec) / KbT)))
	t_1 = NaChar / (1.0 + math.exp((Vef / KbT)))
	tmp = 0
	if Vef <= -9.8e+213:
		tmp = t_1
	elif Vef <= 2.1e-158:
		tmp = t_0
	elif Vef <= 2.35e-114:
		tmp = NaChar / (1.0 + math.exp((-mu / KbT)))
	elif Vef <= 6e+191:
		tmp = t_0
	else:
		tmp = t_1
	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(mu + EDonor) - Ec) / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT))))
	tmp = 0.0
	if (Vef <= -9.8e+213)
		tmp = t_1;
	elseif (Vef <= 2.1e-158)
		tmp = t_0;
	elseif (Vef <= 2.35e-114)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT))));
	elseif (Vef <= 6e+191)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((((mu + EDonor) - Ec) / KbT)));
	t_1 = NaChar / (1.0 + exp((Vef / KbT)));
	tmp = 0.0;
	if (Vef <= -9.8e+213)
		tmp = t_1;
	elseif (Vef <= 2.1e-158)
		tmp = t_0;
	elseif (Vef <= 2.35e-114)
		tmp = NaChar / (1.0 + exp((-mu / KbT)));
	elseif (Vef <= 6e+191)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -9.8e+213], t$95$1, If[LessEqual[Vef, 2.1e-158], t$95$0, If[LessEqual[Vef, 2.35e-114], N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 6e+191], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -9.79999999999999994e213 or 5.9999999999999995e191 < Vef

   1. Initial program 100.0%

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

   3. Taylor expanded in NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 80.2%

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

   if -9.79999999999999994e213 < Vef < 2.09999999999999991e-158 or 2.35000000000000003e-114 < Vef < 5.9999999999999995e191

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 64.2%

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

   if 2.09999999999999991e-158 < Vef < 2.35000000000000003e-114

   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 NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in mu around inf

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

   8. Applied rewrites 83.9%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -9.8 \cdot 10^{+213}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;Vef \leq 2.35 \cdot 10^{-114}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 6 \cdot 10^{+191}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{if}\;NdChar \leq -5.8 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 1.38 \cdot 10^{+109}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))))))
   (if (<= NdChar -5.8e+154)
     t_0
     (if (<= NdChar 1.38e+109)
       (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) 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 = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	double tmp;
	if (NdChar <= -5.8e+154) {
		tmp = t_0;
	} else if (NdChar <= 1.38e+109) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((((mu + vef) + edonor) - ec) / kbt)))
    if (ndchar <= (-5.8d+154)) then
        tmp = t_0
    else if (ndchar <= 1.38d+109) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	double tmp;
	if (NdChar <= -5.8e+154) {
		tmp = t_0;
	} else if (NdChar <= 1.38e+109) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)))
	tmp = 0
	if NdChar <= -5.8e+154:
		tmp = t_0
	elif NdChar <= 1.38e+109:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = t_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(mu + Vef) + EDonor) - Ec) / KbT))))
	tmp = 0.0
	if (NdChar <= -5.8e+154)
		tmp = t_0;
	elseif (NdChar <= 1.38e+109)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	tmp = 0.0;
	if (NdChar <= -5.8e+154)
		tmp = t_0;
	elseif (NdChar <= 1.38e+109)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -5.8e+154], t$95$0, If[LessEqual[NdChar, 1.38e+109], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -5.79999999999999959e154 or 1.37999999999999994e109 < NdChar

   1. Initial program 100.0%

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

   3. Taylor expanded in NaChar around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if -5.79999999999999959e154 < NdChar < 1.37999999999999994e109

   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 NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 71.7

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

   5. Applied rewrites 71.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -5.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.38 \cdot 10^{+109}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{if}\;NdChar \leq -5.8 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 6.9 \cdot 10^{+109}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (+ mu EDonor) Ec) KbT))))))
   (if (<= NdChar -5.8e+154)
     t_0
     (if (<= NdChar 6.9e+109)
       (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) 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 = NdChar / (1.0 + exp((((mu + EDonor) - Ec) / KbT)));
	double tmp;
	if (NdChar <= -5.8e+154) {
		tmp = t_0;
	} else if (NdChar <= 6.9e+109) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((((mu + edonor) - ec) / kbt)))
    if (ndchar <= (-5.8d+154)) then
        tmp = t_0
    else if (ndchar <= 6.9d+109) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((((mu + EDonor) - Ec) / KbT)));
	double tmp;
	if (NdChar <= -5.8e+154) {
		tmp = t_0;
	} else if (NdChar <= 6.9e+109) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((((mu + EDonor) - Ec) / KbT)))
	tmp = 0
	if NdChar <= -5.8e+154:
		tmp = t_0
	elif NdChar <= 6.9e+109:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = t_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(mu + EDonor) - Ec) / KbT))))
	tmp = 0.0
	if (NdChar <= -5.8e+154)
		tmp = t_0;
	elseif (NdChar <= 6.9e+109)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((((mu + EDonor) - Ec) / KbT)));
	tmp = 0.0;
	if (NdChar <= -5.8e+154)
		tmp = t_0;
	elseif (NdChar <= 6.9e+109)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -5.8e+154], t$95$0, If[LessEqual[NdChar, 6.9e+109], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -5.79999999999999959e154 or 6.8999999999999999e109 < NdChar

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 77.2%

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

   if -5.79999999999999959e154 < NdChar < 6.8999999999999999e109

   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 NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 71.7

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

   5. Applied rewrites 71.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -5.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 6.9 \cdot 10^{+109}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{if}\;EDonor \leq -3550:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;EDonor \leq 1.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;EDonor \leq 9.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))))
   (if (<= EDonor -3550.0)
     t_0
     (if (<= EDonor 1.6e-232)
       (/ NdChar (+ 1.0 (exp (/ mu KbT))))
       (if (<= EDonor 9.6e+55)
         (/ NaChar (+ 1.0 (exp (/ EAccept 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 = NdChar / (1.0 + exp((EDonor / KbT)));
	double tmp;
	if (EDonor <= -3550.0) {
		tmp = t_0;
	} else if (EDonor <= 1.6e-232) {
		tmp = NdChar / (1.0 + exp((mu / KbT)));
	} else if (EDonor <= 9.6e+55) {
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((edonor / kbt)))
    if (edonor <= (-3550.0d0)) then
        tmp = t_0
    else if (edonor <= 1.6d-232) then
        tmp = ndchar / (1.0d0 + exp((mu / kbt)))
    else if (edonor <= 9.6d+55) then
        tmp = nachar / (1.0d0 + exp((eaccept / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	double tmp;
	if (EDonor <= -3550.0) {
		tmp = t_0;
	} else if (EDonor <= 1.6e-232) {
		tmp = NdChar / (1.0 + Math.exp((mu / KbT)));
	} else if (EDonor <= 9.6e+55) {
		tmp = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((EDonor / KbT)))
	tmp = 0
	if EDonor <= -3550.0:
		tmp = t_0
	elif EDonor <= 1.6e-232:
		tmp = NdChar / (1.0 + math.exp((mu / KbT)))
	elif EDonor <= 9.6e+55:
		tmp = NaChar / (1.0 + math.exp((EAccept / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))))
	tmp = 0.0
	if (EDonor <= -3550.0)
		tmp = t_0;
	elseif (EDonor <= 1.6e-232)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT))));
	elseif (EDonor <= 9.6e+55)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((EDonor / KbT)));
	tmp = 0.0;
	if (EDonor <= -3550.0)
		tmp = t_0;
	elseif (EDonor <= 1.6e-232)
		tmp = NdChar / (1.0 + exp((mu / KbT)));
	elseif (EDonor <= 9.6e+55)
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EDonor, -3550.0], t$95$0, If[LessEqual[EDonor, 1.6e-232], N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 9.6e+55], N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if EDonor < -3550 or 9.5999999999999997e55 < EDonor

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 59.8%

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

   9. Taylor expanded in EDonor around inf

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

   11. Applied rewrites 53.4%

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

   if -3550 < EDonor < 1.59999999999999993e-232

   1. Initial program 99.7%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 61.9%

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

   9. Taylor expanded in mu around inf

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

   11. Applied rewrites 44.2%

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

   if 1.59999999999999993e-232 < EDonor < 9.5999999999999997e55

   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 NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 42.0%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -3550:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;EDonor \leq 1.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;EDonor \leq 9.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Vef \leq -3.8 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 1.95 \cdot 10^{+46}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ Vef KbT))))))
   (if (<= Vef -3.8e+96)
     t_0
     (if (<= Vef 1.95e+46) (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((Vef / KbT)));
	double tmp;
	if (Vef <= -3.8e+96) {
		tmp = t_0;
	} else if (Vef <= 1.95e+46) {
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((vef / kbt)))
    if (vef <= (-3.8d+96)) then
        tmp = t_0
    else if (vef <= 1.95d+46) then
        tmp = ndchar / (1.0d0 + exp((edonor / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((Vef / KbT)));
	double tmp;
	if (Vef <= -3.8e+96) {
		tmp = t_0;
	} else if (Vef <= 1.95e+46) {
		tmp = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((Vef / KbT)))
	tmp = 0
	if Vef <= -3.8e+96:
		tmp = t_0
	elif Vef <= 1.95e+46:
		tmp = NdChar / (1.0 + math.exp((EDonor / KbT)))
	else:
		tmp = 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(Vef / KbT))))
	tmp = 0.0
	if (Vef <= -3.8e+96)
		tmp = t_0;
	elseif (Vef <= 1.95e+46)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((Vef / KbT)));
	tmp = 0.0;
	if (Vef <= -3.8e+96)
		tmp = t_0;
	elseif (Vef <= 1.95e+46)
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -3.8e+96], t$95$0, If[LessEqual[Vef, 1.95e+46], N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -3.8000000000000002e96 or 1.94999999999999997e46 < Vef

   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 NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 59.8%

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

   if -3.8000000000000002e96 < Vef < 1.94999999999999997e46

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 63.3%

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

   9. Taylor expanded in EDonor around inf

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

   11. Applied rewrites 45.0%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -3.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.95 \cdot 10^{+46}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 40.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\\ \mathbf{if}\;KbT \leq -1.45 \cdot 10^{+144}:\\ \;\;\;\;0.5 \cdot NaChar + \mathsf{fma}\left(-0.25, t\_0 \cdot NdChar, 0.5 \cdot NdChar\right)\\ \mathbf{elif}\;KbT \leq 2.4 \cdot 10^{+191}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\mathsf{fma}\left(1, t\_0, 2\right)} - \frac{NaChar}{-1 - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ (- (+ (+ mu Vef) EDonor) Ec) KbT)))
   (if (<= KbT -1.45e+144)
     (+ (* 0.5 NaChar) (fma -0.25 (* t_0 NdChar) (* 0.5 NdChar)))
     (if (<= KbT 2.4e+191)
       (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
       (- (/ NdChar (fma 1.0 t_0 2.0)) (/ NaChar (- -1.0 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 = (((mu + Vef) + EDonor) - Ec) / KbT;
	double tmp;
	if (KbT <= -1.45e+144) {
		tmp = (0.5 * NaChar) + fma(-0.25, (t_0 * NdChar), (0.5 * NdChar));
	} else if (KbT <= 2.4e+191) {
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	} else {
		tmp = (NdChar / fma(1.0, t_0, 2.0)) - (NaChar / (-1.0 - 1.0));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT)
	tmp = 0.0
	if (KbT <= -1.45e+144)
		tmp = Float64(Float64(0.5 * NaChar) + fma(-0.25, Float64(t_0 * NdChar), Float64(0.5 * NdChar)));
	elseif (KbT <= 2.4e+191)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))));
	else
		tmp = Float64(Float64(NdChar / fma(1.0, t_0, 2.0)) - Float64(NaChar / Float64(-1.0 - 1.0)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]}, If[LessEqual[KbT, -1.45e+144], N[(N[(0.5 * NaChar), $MachinePrecision] + N[(-0.25 * N[(t$95$0 * NdChar), $MachinePrecision] + N[(0.5 * NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.4e+191], N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 * t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.44999999999999999e144

   1. Initial program 99.7%

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

   3. Taylor expanded in KbT around inf

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 49.2

          \[\leadsto \mathsf{fma}\left(-0.25, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}, \color{blue}{0.5 \cdot NdChar}\right) + \frac{NaChar}{1 + 1} \]

   8. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}, 0.5 \cdot NdChar\right)} + \frac{NaChar}{1 + 1} \]

   9. Taylor expanded in KbT around inf

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

       1. lower-*.f64 49.2

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

   11. Applied rewrites 49.2%

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

   if -1.44999999999999999e144 < KbT < 2.39999999999999986e191

   1. Initial program 100.0%

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

   3. Taylor expanded in Vef around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in EDonor around inf

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

   11. Applied rewrites 41.0%

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

   if 2.39999999999999986e191 < KbT

   1. Initial program 99.7%

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

   3. Taylor expanded in KbT around inf

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

   5. Applied rewrites 84.2%

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

      1. lift-exp.f64 N/A

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

      2. *-lft-identity N/A

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

      3. pow-exp N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-pow.f64 84.4

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

      6. lift-exp.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 84.4

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in KbT around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. log-E N/A

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

      7. mul-1-neg N/A

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

      8. associate-+r+ N/A

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

      9. sub-neg N/A

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

      10. associate--l+ N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 74.1%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.45 \cdot 10^{+144}:\\ \;\;\;\;0.5 \cdot NaChar + \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, 0.5 \cdot NdChar\right)\\ \mathbf{elif}\;KbT \leq 2.4 \cdot 10^{+191}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\mathsf{fma}\left(1, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}, 2\right)} - \frac{NaChar}{-1 - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 22.7% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -4.1 \cdot 10^{-63}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{elif}\;NdChar \leq 3.6 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -4.1e-63)
   (* 0.5 NdChar)
   (if (<= NdChar 3.6e-19) (* 0.5 NaChar) (* 0.5 NdChar))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -4.1e-63) {
		tmp = 0.5 * NdChar;
	} else if (NdChar <= 3.6e-19) {
		tmp = 0.5 * NaChar;
	} else {
		tmp = 0.5 * NdChar;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-4.1d-63)) then
        tmp = 0.5d0 * ndchar
    else if (ndchar <= 3.6d-19) then
        tmp = 0.5d0 * nachar
    else
        tmp = 0.5d0 * ndchar
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -4.1e-63) {
		tmp = 0.5 * NdChar;
	} else if (NdChar <= 3.6e-19) {
		tmp = 0.5 * NaChar;
	} else {
		tmp = 0.5 * NdChar;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -4.1e-63:
		tmp = 0.5 * NdChar
	elif NdChar <= 3.6e-19:
		tmp = 0.5 * NaChar
	else:
		tmp = 0.5 * NdChar
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -4.1e-63)
		tmp = Float64(0.5 * NdChar);
	elseif (NdChar <= 3.6e-19)
		tmp = Float64(0.5 * NaChar);
	else
		tmp = Float64(0.5 * NdChar);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -4.1e-63)
		tmp = 0.5 * NdChar;
	elseif (NdChar <= 3.6e-19)
		tmp = 0.5 * NaChar;
	else
		tmp = 0.5 * NdChar;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -4.1e-63], N[(0.5 * NdChar), $MachinePrecision], If[LessEqual[NdChar, 3.6e-19], N[(0.5 * NaChar), $MachinePrecision], N[(0.5 * NdChar), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -4.0999999999999998e-63 or 3.6000000000000001e-19 < NdChar

   1. Initial program 100.0%

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

   3. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 27.0

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

   5. Applied rewrites 27.0%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 24.2%

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

   if -4.0999999999999998e-63 < NdChar < 3.6000000000000001e-19

   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 NaChar around inf

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   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. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 25.3%

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

Alternative 19: 27.8% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 26.6

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

5. Applied rewrites 26.6%

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

6. Final simplification 26.6%

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

Alternative 20: 18.0% 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 99.9%

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

3. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 26.6

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

5. Applied rewrites 26.6%

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

6. Taylor expanded in NaChar around 0

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

8. Applied rewrites 18.3%

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

Reproduce

herbie shell --seed 2024271 
(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))))))