Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 16.0s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 8.281270336300764e+242
  2. Ec = 3.235159701412171e-209
  3. Vef = -3.810753841166931e+289
  4. EDonor = 8.812601612884842e-163
  5. mu = 2.8740647259041523e+142
  6. KbT = 1.1998574507597012e+190
  7. NaChar = -3.686418252925923e+243
  8. Ev = 6.196170125260445e-100
  9. EAccept = -7.151480850421675e+295

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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 2: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - t\_0}\\ t_2 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_3 := \frac{NdChar}{1 + t\_0} - \frac{NaChar}{-1 - t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\frac{NaChar}{1 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)))
        (t_1 (- (/ NaChar (+ 1.0 (exp (/ Vef KbT)))) (/ NdChar (- -1.0 t_0))))
        (t_2 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_3 (- (/ NdChar (+ 1.0 t_0)) (/ NaChar (- -1.0 t_2)))))
   (if (<= t_3 -2e-120)
     t_1
     (if (<= t_3 3.7e-263) (/ NaChar (+ 1.0 t_2)) t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	double t_1 = (NaChar / (1.0 + exp((Vef / KbT)))) - (NdChar / (-1.0 - t_0));
	double t_2 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_3 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - t_2));
	double tmp;
	if (t_3 <= -2e-120) {
		tmp = t_1;
	} else if (t_3 <= 3.7e-263) {
		tmp = NaChar / (1.0 + t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = exp(((mu - ((ec - vef) - edonor)) / kbt))
    t_1 = (nachar / (1.0d0 + exp((vef / kbt)))) - (ndchar / ((-1.0d0) - t_0))
    t_2 = exp((((eaccept + (ev + vef)) - mu) / kbt))
    t_3 = (ndchar / (1.0d0 + t_0)) - (nachar / ((-1.0d0) - t_2))
    if (t_3 <= (-2d-120)) then
        tmp = t_1
    else if (t_3 <= 3.7d-263) then
        tmp = nachar / (1.0d0 + t_2)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	double t_1 = (NaChar / (1.0 + Math.exp((Vef / KbT)))) - (NdChar / (-1.0 - t_0));
	double t_2 = Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_3 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - t_2));
	double tmp;
	if (t_3 <= -2e-120) {
		tmp = t_1;
	} else if (t_3 <= 3.7e-263) {
		tmp = NaChar / (1.0 + t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))
	t_1 = (NaChar / (1.0 + math.exp((Vef / KbT)))) - (NdChar / (-1.0 - t_0))
	t_2 = math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))
	t_3 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - t_2))
	tmp = 0
	if t_3 <= -2e-120:
		tmp = t_1
	elif t_3 <= 3.7e-263:
		tmp = NaChar / (1.0 + t_2)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) - Float64(NdChar / Float64(-1.0 - t_0)))
	t_2 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_3 = Float64(Float64(NdChar / Float64(1.0 + t_0)) - Float64(NaChar / Float64(-1.0 - t_2)))
	tmp = 0.0
	if (t_3 <= -2e-120)
		tmp = t_1;
	elseif (t_3 <= 3.7e-263)
		tmp = Float64(NaChar / Float64(1.0 + t_2));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	t_1 = (NaChar / (1.0 + exp((Vef / KbT)))) - (NdChar / (-1.0 - t_0));
	t_2 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	t_3 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - t_2));
	tmp = 0.0;
	if (t_3 <= -2e-120)
		tmp = t_1;
	elseif (t_3 <= 3.7e-263)
		tmp = NaChar / (1.0 + t_2);
	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[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-120], t$95$1, If[LessEqual[t$95$3, 3.7e-263], N[(NaChar / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.99999999999999996e-120 or 3.6999999999999997e-263 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in Vef around inf

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

      1. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if -1.99999999999999996e-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))))) < 3.6999999999999997e-263

   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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -2 \cdot 10^{-120}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{t\_1}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-237}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\frac{NaChar \cdot NdChar}{\mathsf{fma}\left(2, NdChar, t\_1 \cdot NdChar\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1 (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))
        (t_2
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp t_1))))))
   (if (<= t_2 -1e-237)
     t_0
     (if (<= t_2 3.7e-263)
       (/ (* NaChar NdChar) (fma 2.0 NdChar (* t_1 NdChar)))
       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 = ((EAccept + (Ev + Vef)) - mu) / KbT;
	double t_2 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp(t_1)));
	double tmp;
	if (t_2 <= -1e-237) {
		tmp = t_0;
	} else if (t_2 <= 3.7e-263) {
		tmp = (NaChar * NdChar) / fma(2.0, NdChar, (t_1 * NdChar));
	} 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(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 - exp(t_1))))
	tmp = 0.0
	if (t_2 <= -1e-237)
		tmp = t_0;
	elseif (t_2 <= 3.7e-263)
		tmp = Float64(Float64(NaChar * NdChar) / fma(2.0, NdChar, Float64(t_1 * NdChar)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $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 - N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-237], t$95$0, If[LessEqual[t$95$2, 3.7e-263], N[(N[(NaChar * NdChar), $MachinePrecision] / N[(2.0 * NdChar + N[(t$95$1 * NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999999e-238 or 3.6999999999999997e-263 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   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 40.9

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

   5. Applied rewrites 40.9%

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

   if -9.9999999999999999e-238 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 3.6999999999999997e-263

   1. Initial program 100.0%

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

   3. Taylor expanded in NdChar around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in NaChar around inf

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

   8. Applied rewrites 76.3%

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

   9. Taylor expanded in KbT around inf

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

   11. Applied rewrites 52.3%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -1 \cdot 10^{-237}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\frac{NaChar \cdot NdChar}{\mathsf{fma}\left(2, NdChar, \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot NdChar\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-237}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\frac{1}{\frac{1}{NaChar} - \frac{NdChar}{NaChar \cdot NaChar}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ 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 -1e-237)
     t_0
     (if (<= t_1 3.7e-263)
       (* (/ 1.0 (- (/ 1.0 NaChar) (/ NdChar (* NaChar NaChar)))) 0.5)
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (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 <= -1e-237) {
		tmp = t_0;
	} else if (t_1 <= 3.7e-263) {
		tmp = (1.0 / ((1.0 / NaChar) - (NdChar / (NaChar * NaChar)))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (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 <= (-1d-237)) then
        tmp = t_0
    else if (t_1 <= 3.7d-263) then
        tmp = (1.0d0 / ((1.0d0 / nachar) - (ndchar / (nachar * nachar)))) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (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 <= -1e-237) {
		tmp = t_0;
	} else if (t_1 <= 3.7e-263) {
		tmp = (1.0 / ((1.0 / NaChar) - (NdChar / (NaChar * NaChar)))) * 0.5;
	} 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 <= -1e-237:
		tmp = t_0
	elif t_1 <= 3.7e-263:
		tmp = (1.0 / ((1.0 / NaChar) - (NdChar / (NaChar * NaChar)))) * 0.5
	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 <= -1e-237)
		tmp = t_0;
	elseif (t_1 <= 3.7e-263)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 / NaChar) - Float64(NdChar / Float64(NaChar * NaChar)))) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (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 <= -1e-237)
		tmp = t_0;
	elseif (t_1 <= 3.7e-263)
		tmp = (1.0 / ((1.0 / NaChar) - (NdChar / (NaChar * NaChar)))) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(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, -1e-237], t$95$0, If[LessEqual[t$95$1, 3.7e-263], N[(N[(1.0 / N[(N[(1.0 / NaChar), $MachinePrecision] - N[(NdChar / N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999999e-238 or 3.6999999999999997e-263 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   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 40.9

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

   5. Applied rewrites 40.9%

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

   if -9.9999999999999999e-238 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 3.6999999999999997e-263

   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 2.7

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in NdChar around 0

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

   10. Applied rewrites 28.4%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -1 \cdot 10^{-237}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\frac{1}{\frac{1}{NaChar} - \frac{NdChar}{NaChar \cdot NaChar}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-238}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\frac{1}{\left(NaChar + NdChar\right) \cdot \frac{1}{NaChar \cdot NaChar}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ 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 -1e-238)
     t_0
     (if (<= t_1 3.7e-263)
       (* (/ 1.0 (* (+ NaChar NdChar) (/ 1.0 (* NaChar NaChar)))) 0.5)
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (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 <= -1e-238) {
		tmp = t_0;
	} else if (t_1 <= 3.7e-263) {
		tmp = (1.0 / ((NaChar + NdChar) * (1.0 / (NaChar * NaChar)))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (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 <= (-1d-238)) then
        tmp = t_0
    else if (t_1 <= 3.7d-263) then
        tmp = (1.0d0 / ((nachar + ndchar) * (1.0d0 / (nachar * nachar)))) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (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 <= -1e-238) {
		tmp = t_0;
	} else if (t_1 <= 3.7e-263) {
		tmp = (1.0 / ((NaChar + NdChar) * (1.0 / (NaChar * NaChar)))) * 0.5;
	} 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 <= -1e-238:
		tmp = t_0
	elif t_1 <= 3.7e-263:
		tmp = (1.0 / ((NaChar + NdChar) * (1.0 / (NaChar * NaChar)))) * 0.5
	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 <= -1e-238)
		tmp = t_0;
	elseif (t_1 <= 3.7e-263)
		tmp = Float64(Float64(1.0 / Float64(Float64(NaChar + NdChar) * Float64(1.0 / Float64(NaChar * NaChar)))) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (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 <= -1e-238)
		tmp = t_0;
	elseif (t_1 <= 3.7e-263)
		tmp = (1.0 / ((NaChar + NdChar) * (1.0 / (NaChar * NaChar)))) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(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, -1e-238], t$95$0, If[LessEqual[t$95$1, 3.7e-263], N[(N[(1.0 / N[(N[(NaChar + NdChar), $MachinePrecision] * N[(1.0 / N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999999e-239 or 3.6999999999999997e-263 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   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 40.7

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

   5. Applied rewrites 40.7%

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

   if -9.9999999999999999e-239 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 3.6999999999999997e-263

   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 2.7

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 4.2%

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

   9. Applied rewrites 4.2%

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

   10. Taylor expanded in NaChar around inf

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

   12. Applied rewrites 28.7%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -1 \cdot 10^{-238}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\frac{1}{\left(NaChar + NdChar\right) \cdot \frac{1}{NaChar \cdot NaChar}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\left(\frac{NaChar}{KbT} \cdot Ev\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1e-240)
     t_0
     (if (<= t_1 3.7e-263) (* (* (/ NaChar KbT) Ev) -0.25) 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 <= -1e-240) {
		tmp = t_0;
	} else if (t_1 <= 3.7e-263) {
		tmp = ((NaChar / KbT) * Ev) * -0.25;
	} 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 <= (-1d-240)) then
        tmp = t_0
    else if (t_1 <= 3.7d-263) then
        tmp = ((nachar / kbt) * ev) * (-0.25d0)
    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 <= -1e-240) {
		tmp = t_0;
	} else if (t_1 <= 3.7e-263) {
		tmp = ((NaChar / KbT) * Ev) * -0.25;
	} 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 <= -1e-240:
		tmp = t_0
	elif t_1 <= 3.7e-263:
		tmp = ((NaChar / KbT) * Ev) * -0.25
	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 <= -1e-240)
		tmp = t_0;
	elseif (t_1 <= 3.7e-263)
		tmp = Float64(Float64(Float64(NaChar / KbT) * Ev) * -0.25);
	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 <= -1e-240)
		tmp = t_0;
	elseif (t_1 <= 3.7e-263)
		tmp = ((NaChar / KbT) * Ev) * -0.25;
	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, -1e-240], t$95$0, If[LessEqual[t$95$1, 3.7e-263], N[(N[(N[(NaChar / KbT), $MachinePrecision] * Ev), $MachinePrecision] * -0.25), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.9999999999999997e-241 or 3.6999999999999997e-263 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   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 40.5

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

   5. Applied rewrites 40.5%

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

   if -9.9999999999999997e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 3.6999999999999997e-263

   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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in Ev around inf

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

   8. Applied rewrites 40.6%

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

   9. Taylor expanded in KbT around inf

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

   11. Applied rewrites 4.7%

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

   12. Taylor expanded in Ev around inf

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

   14. Applied rewrites 13.9%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -1 \cdot 10^{-240}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 3.7 \cdot 10^{-263}:\\ \;\;\;\;\left(\frac{NaChar}{KbT} \cdot Ev\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ t_1 := \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar\\ \mathbf{if}\;KbT \leq -6 \cdot 10^{+92}:\\ \;\;\;\;\frac{NaChar}{2} + \mathsf{fma}\left(-0.25, t\_1, 0.5 \cdot NdChar\right)\\ \mathbf{elif}\;KbT \leq -3.9 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 4 \cdot 10^{-35}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot NaChar, \mathsf{fma}\left(-0.25, t\_1, \left(NaChar + NdChar\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
        (t_1 (* (/ (- (+ (+ mu Vef) EDonor) Ec) KbT) NdChar)))
   (if (<= KbT -6e+92)
     (+ (/ NaChar 2.0) (fma -0.25 t_1 (* 0.5 NdChar)))
     (if (<= KbT -3.9e-99)
       t_0
       (if (<= KbT 4e-35)
         (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))
         (if (<= KbT 8.2e+116)
           t_0
           (fma
            -0.25
            (* (/ (- (+ EAccept (+ Ev Vef)) mu) KbT) NaChar)
            (fma -0.25 t_1 (* (+ 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) {
	double t_0 = NaChar / (1.0 + exp((Ev / KbT)));
	double t_1 = ((((mu + Vef) + EDonor) - Ec) / KbT) * NdChar;
	double tmp;
	if (KbT <= -6e+92) {
		tmp = (NaChar / 2.0) + fma(-0.25, t_1, (0.5 * NdChar));
	} else if (KbT <= -3.9e-99) {
		tmp = t_0;
	} else if (KbT <= 4e-35) {
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	} else if (KbT <= 8.2e+116) {
		tmp = t_0;
	} else {
		tmp = fma(-0.25, ((((EAccept + (Ev + Vef)) - mu) / KbT) * NaChar), fma(-0.25, t_1, ((NaChar + NdChar) * 0.5)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))))
	t_1 = Float64(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT) * NdChar)
	tmp = 0.0
	if (KbT <= -6e+92)
		tmp = Float64(Float64(NaChar / 2.0) + fma(-0.25, t_1, Float64(0.5 * NdChar)));
	elseif (KbT <= -3.9e-99)
		tmp = t_0;
	elseif (KbT <= 4e-35)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))));
	elseif (KbT <= 8.2e+116)
		tmp = t_0;
	else
		tmp = fma(-0.25, Float64(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT) * NaChar), fma(-0.25, t_1, Float64(Float64(NaChar + NdChar) * 0.5)));
	end
	return 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[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision] * NdChar), $MachinePrecision]}, If[LessEqual[KbT, -6e+92], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(-0.25 * t$95$1 + N[(0.5 * NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -3.9e-99], t$95$0, If[LessEqual[KbT, 4e-35], N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 8.2e+116], t$95$0, N[(-0.25 * N[(N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision] * NaChar), $MachinePrecision] + N[(-0.25 * t$95$1 + N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -6.00000000000000026e92

   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

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

      1. lower-fma.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 63.2%

      \[\leadsto \mathsf{fma}\left(-0.25, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}, 0.5 \cdot NdChar\right) + \frac{NaChar}{\color{blue}{2}} \]

   if -6.00000000000000026e92 < KbT < -3.89999999999999987e-99 or 4.00000000000000003e-35 < KbT < 8.1999999999999996e116

   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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in Ev around inf

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

   8. Applied rewrites 30.6%

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

   if -3.89999999999999987e-99 < KbT < 4.00000000000000003e-35

   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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 33.4%

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

   if 8.1999999999999996e116 < KbT

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. lower-fma.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, NaChar \cdot \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}, \mathsf{fma}\left(-0.25, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}, 0.5 \cdot \left(NaChar + NdChar\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6 \cdot 10^{+92}:\\ \;\;\;\;\frac{NaChar}{2} + \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, 0.5 \cdot NdChar\right)\\ \mathbf{elif}\;KbT \leq -3.9 \cdot 10^{-99}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;KbT \leq 4 \cdot 10^{-35}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot NaChar, \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, \left(NaChar + NdChar\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{if}\;NdChar \leq -4.7 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 5.4 \cdot 10^{+35}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -4.7e-136)
     t_0
     (if (<= NdChar 5.4e+35)
       (/ 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 <= -4.7e-136) {
		tmp = t_0;
	} else if (NdChar <= 5.4e+35) {
		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 <= (-4.7d-136)) then
        tmp = t_0
    else if (ndchar <= 5.4d+35) 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 <= -4.7e-136) {
		tmp = t_0;
	} else if (NdChar <= 5.4e+35) {
		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 <= -4.7e-136:
		tmp = t_0
	elif NdChar <= 5.4e+35:
		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 <= -4.7e-136)
		tmp = t_0;
	elseif (NdChar <= 5.4e+35)
		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 <= -4.7e-136)
		tmp = t_0;
	elseif (NdChar <= 5.4e+35)
		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, -4.7e-136], t$95$0, If[LessEqual[NdChar, 5.4e+35], 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 < -4.70000000000000022e-136 or 5.40000000000000005e35 < NdChar

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 73.9

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

   5. Applied rewrites 73.9%

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

   if -4.70000000000000022e-136 < NdChar < 5.40000000000000005e35

   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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 73.5

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

   5. Applied rewrites 73.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -4.7 \cdot 10^{-136}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 5.4 \cdot 10^{+35}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar\\ t_1 := \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\\ \mathbf{if}\;KbT \leq -7.4 \cdot 10^{+178}:\\ \;\;\;\;\frac{NaChar}{2} + \mathsf{fma}\left(-0.25, t\_0, 0.5 \cdot NdChar\right)\\ \mathbf{elif}\;KbT \leq 1.82 \cdot 10^{+149}:\\ \;\;\;\;\frac{NaChar}{1 + e^{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, t\_1 \cdot NaChar, \mathsf{fma}\left(-0.25, t\_0, \left(NaChar + NdChar\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (/ (- (+ (+ mu Vef) EDonor) Ec) KbT) NdChar))
        (t_1 (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
   (if (<= KbT -7.4e+178)
     (+ (/ NaChar 2.0) (fma -0.25 t_0 (* 0.5 NdChar)))
     (if (<= KbT 1.82e+149)
       (/ NaChar (+ 1.0 (exp t_1)))
       (fma -0.25 (* t_1 NaChar) (fma -0.25 t_0 (* (+ 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) {
	double t_0 = ((((mu + Vef) + EDonor) - Ec) / KbT) * NdChar;
	double t_1 = ((EAccept + (Ev + Vef)) - mu) / KbT;
	double tmp;
	if (KbT <= -7.4e+178) {
		tmp = (NaChar / 2.0) + fma(-0.25, t_0, (0.5 * NdChar));
	} else if (KbT <= 1.82e+149) {
		tmp = NaChar / (1.0 + exp(t_1));
	} else {
		tmp = fma(-0.25, (t_1 * NaChar), fma(-0.25, t_0, ((NaChar + NdChar) * 0.5)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT) * NdChar)
	t_1 = Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)
	tmp = 0.0
	if (KbT <= -7.4e+178)
		tmp = Float64(Float64(NaChar / 2.0) + fma(-0.25, t_0, Float64(0.5 * NdChar)));
	elseif (KbT <= 1.82e+149)
		tmp = Float64(NaChar / Float64(1.0 + exp(t_1)));
	else
		tmp = fma(-0.25, Float64(t_1 * NaChar), fma(-0.25, t_0, Float64(Float64(NaChar + NdChar) * 0.5)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision] * NdChar), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]}, If[LessEqual[KbT, -7.4e+178], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(-0.25 * t$95$0 + N[(0.5 * NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.82e+149], N[(NaChar / N[(1.0 + N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(t$95$1 * NaChar), $MachinePrecision] + N[(-0.25 * t$95$0 + N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -7.4000000000000005e178

   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

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

      1. lower-fma.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 82.2%

      \[\leadsto \mathsf{fma}\left(-0.25, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}, 0.5 \cdot NdChar\right) + \frac{NaChar}{\color{blue}{2}} \]

   if -7.4000000000000005e178 < KbT < 1.8199999999999999e149

   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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if 1.8199999999999999e149 < KbT

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. lower-fma.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 72.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -7.4 \cdot 10^{+178}:\\ \;\;\;\;\frac{NaChar}{2} + \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, 0.5 \cdot NdChar\right)\\ \mathbf{elif}\;KbT \leq 1.82 \cdot 10^{+149}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot NaChar, \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, \left(NaChar + NdChar\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar\\ \mathbf{if}\;KbT \leq -6 \cdot 10^{+92}:\\ \;\;\;\;\frac{NaChar}{2} + \mathsf{fma}\left(-0.25, t\_0, 0.5 \cdot NdChar\right)\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot NaChar, \mathsf{fma}\left(-0.25, t\_0, \left(NaChar + NdChar\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (/ (- (+ (+ mu Vef) EDonor) Ec) KbT) NdChar)))
   (if (<= KbT -6e+92)
     (+ (/ NaChar 2.0) (fma -0.25 t_0 (* 0.5 NdChar)))
     (if (<= KbT 2.6e+117)
       (/ NaChar (+ 1.0 (exp (/ Vef KbT))))
       (fma
        -0.25
        (* (/ (- (+ EAccept (+ Ev Vef)) mu) KbT) NaChar)
        (fma -0.25 t_0 (* (+ 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) {
	double t_0 = ((((mu + Vef) + EDonor) - Ec) / KbT) * NdChar;
	double tmp;
	if (KbT <= -6e+92) {
		tmp = (NaChar / 2.0) + fma(-0.25, t_0, (0.5 * NdChar));
	} else if (KbT <= 2.6e+117) {
		tmp = NaChar / (1.0 + exp((Vef / KbT)));
	} else {
		tmp = fma(-0.25, ((((EAccept + (Ev + Vef)) - mu) / KbT) * NaChar), fma(-0.25, t_0, ((NaChar + NdChar) * 0.5)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT) * NdChar)
	tmp = 0.0
	if (KbT <= -6e+92)
		tmp = Float64(Float64(NaChar / 2.0) + fma(-0.25, t_0, Float64(0.5 * NdChar)));
	elseif (KbT <= 2.6e+117)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT))));
	else
		tmp = fma(-0.25, Float64(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT) * NaChar), fma(-0.25, t_0, Float64(Float64(NaChar + NdChar) * 0.5)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision] * NdChar), $MachinePrecision]}, If[LessEqual[KbT, -6e+92], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(-0.25 * t$95$0 + N[(0.5 * NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.6e+117], N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision] * NaChar), $MachinePrecision] + N[(-0.25 * t$95$0 + N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -6.00000000000000026e92

   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

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

      1. lower-fma.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 63.2%

      \[\leadsto \mathsf{fma}\left(-0.25, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}, 0.5 \cdot NdChar\right) + \frac{NaChar}{\color{blue}{2}} \]

   if -6.00000000000000026e92 < KbT < 2.5999999999999999e117

   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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in Vef around inf

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

   8. Applied rewrites 43.9%

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

   if 2.5999999999999999e117 < KbT

   1. Initial program 99.8%

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

   3. Taylor expanded in KbT around -inf

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

      1. lower-fma.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 66.7%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6 \cdot 10^{+92}:\\ \;\;\;\;\frac{NaChar}{2} + \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, 0.5 \cdot NdChar\right)\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot NaChar, \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, \left(NaChar + NdChar\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{2} + \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, 0.5 \cdot NdChar\right)\\ \mathbf{if}\;KbT \leq -3.2 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar 2.0)
          (fma
           -0.25
           (* (/ (- (+ (+ mu Vef) EDonor) Ec) KbT) NdChar)
           (* 0.5 NdChar)))))
   (if (<= KbT -3.2e+123)
     t_0
     (if (<= KbT 7.2e+109) (/ 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 = (NaChar / 2.0) + fma(-0.25, (((((mu + Vef) + EDonor) - Ec) / KbT) * NdChar), (0.5 * NdChar));
	double tmp;
	if (KbT <= -3.2e+123) {
		tmp = t_0;
	} else if (KbT <= 7.2e+109) {
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / 2.0) + fma(-0.25, Float64(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT) * NdChar), Float64(0.5 * NdChar)))
	tmp = 0.0
	if (KbT <= -3.2e+123)
		tmp = t_0;
	elseif (KbT <= 7.2e+109)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / 2.0), $MachinePrecision] + N[(-0.25 * N[(N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision] * NdChar), $MachinePrecision] + N[(0.5 * NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -3.2e+123], t$95$0, If[LessEqual[KbT, 7.2e+109], N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -3.20000000000000005e123 or 7.2e109 < 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

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

      1. lower-fma.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(-0.25, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}, 0.5 \cdot NdChar\right) + \frac{NaChar}{\color{blue}{2}} \]

   if -3.20000000000000005e123 < KbT < 7.2e109

   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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in EAccept around inf

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

   8. Applied rewrites 31.2%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{NaChar}{2} + \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, 0.5 \cdot NdChar\right)\\ \mathbf{elif}\;KbT \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \mathsf{fma}\left(-0.25, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar, 0.5 \cdot NdChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 22.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -5 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{elif}\;NdChar \leq 1.65 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -5e-141)
   (* 0.5 NdChar)
   (if (<= NdChar 1.65e-22) (* 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 <= -5e-141) {
		tmp = 0.5 * NdChar;
	} else if (NdChar <= 1.65e-22) {
		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 <= (-5d-141)) then
        tmp = 0.5d0 * ndchar
    else if (ndchar <= 1.65d-22) 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 <= -5e-141) {
		tmp = 0.5 * NdChar;
	} else if (NdChar <= 1.65e-22) {
		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 <= -5e-141:
		tmp = 0.5 * NdChar
	elif NdChar <= 1.65e-22:
		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 <= -5e-141)
		tmp = Float64(0.5 * NdChar);
	elseif (NdChar <= 1.65e-22)
		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 <= -5e-141)
		tmp = 0.5 * NdChar;
	elseif (NdChar <= 1.65e-22)
		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, -5e-141], N[(0.5 * NdChar), $MachinePrecision], If[LessEqual[NdChar, 1.65e-22], N[(0.5 * NaChar), $MachinePrecision], N[(0.5 * NdChar), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -4.9999999999999999e-141 or 1.65e-22 < NdChar

   1. Initial program 99.9%

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

   3. Taylor expanded in KbT around inf

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

      1. 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 29.6

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in NaChar around 0

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

   8. Applied rewrites 26.7%

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

   if -4.9999999999999999e-141 < NdChar < 1.65e-22

   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. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in KbT around inf

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

   8. Applied rewrites 28.5%

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

Alternative 13: 27.3% 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 30.8

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

5. Applied rewrites 30.8%

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

6. Final simplification 30.8%

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

Alternative 14: 18.5% 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 30.8

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

5. Applied rewrites 30.8%

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

6. Taylor expanded in NaChar around 0

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

8. Applied rewrites 21.8%

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

Reproduce

herbie shell --seed 2024276 
(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))))))