Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 22

Speedup: N/A×

Specification

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    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

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 3: 81.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(Ev, EAccept\right) + Vef\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_2 := \frac{NaChar}{1 + e^{\frac{\left(t\_0 + \mathsf{max}\left(Ev, EAccept\right)\right) + \left(-mu\right)}{KbT}}}\\ t_3 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + t\_2\\ t_4 := t\_1 + t\_2\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(Ec - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\mathsf{min}\left(Ev, EAccept\right)}{KbT}}}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-89}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-221}:\\ \;\;\;\;t\_1 + \frac{NaChar}{1 + e^{\frac{\mathsf{max}\left(Ev, EAccept\right)}{KbT}}}\\ \mathbf{elif}\;t\_4 \leq 10^{-296}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\mathsf{max}\left(Ev, EAccept\right) + t\_0\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (fmin Ev EAccept) Vef))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_2
         (/ NaChar (+ 1.0 (exp (/ (+ (+ t_0 (fmax Ev EAccept)) (- mu)) KbT)))))
        (t_3 (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) t_2))
        (t_4 (+ t_1 t_2)))
   (if (<= t_4 -2e+162)
     (+
      (/ NdChar (+ 1.0 (exp (/ (- (- (- Ec EDonor) mu)) KbT))))
      (/ NaChar (+ 1.0 (exp (/ (fmin Ev EAccept) KbT)))))
     (if (<= t_4 -2e-89)
       t_3
       (if (<= t_4 -5e-221)
         (+ t_1 (/ NaChar (+ 1.0 (exp (/ (fmax Ev EAccept) KbT)))))
         (if (<= t_4 1e-296)
           (/ NaChar (- (exp (/ (- (+ (fmax Ev EAccept) t_0) mu) KbT)) -1.0))
           t_3))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_2 = NaChar / (1.0 + exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT)));
	double t_3 = (NdChar / (1.0 + exp((mu / KbT)))) + t_2;
	double t_4 = t_1 + t_2;
	double tmp;
	if (t_4 <= -2e+162) {
		tmp = (NdChar / (1.0 + exp((-((Ec - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((fmin(Ev, EAccept) / KbT))));
	} else if (t_4 <= -2e-89) {
		tmp = t_3;
	} else if (t_4 <= -5e-221) {
		tmp = t_1 + (NaChar / (1.0 + exp((fmax(Ev, EAccept) / KbT))));
	} else if (t_4 <= 1e-296) {
		tmp = NaChar / (exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    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) :: t_4
    real(8) :: tmp
    t_0 = fmin(ev, eaccept) + vef
    t_1 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_2 = nachar / (1.0d0 + exp((((t_0 + fmax(ev, eaccept)) + -mu) / kbt)))
    t_3 = (ndchar / (1.0d0 + exp((mu / kbt)))) + t_2
    t_4 = t_1 + t_2
    if (t_4 <= (-2d+162)) then
        tmp = (ndchar / (1.0d0 + exp((-((ec - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp((fmin(ev, eaccept) / kbt))))
    else if (t_4 <= (-2d-89)) then
        tmp = t_3
    else if (t_4 <= (-5d-221)) then
        tmp = t_1 + (nachar / (1.0d0 + exp((fmax(ev, eaccept) / kbt))))
    else if (t_4 <= 1d-296) then
        tmp = nachar / (exp((((fmax(ev, eaccept) + t_0) - mu) / kbt)) - (-1.0d0))
    else
        tmp = t_3
    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 = fmin(Ev, EAccept) + Vef;
	double t_1 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_2 = NaChar / (1.0 + Math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT)));
	double t_3 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + t_2;
	double t_4 = t_1 + t_2;
	double tmp;
	if (t_4 <= -2e+162) {
		tmp = (NdChar / (1.0 + Math.exp((-((Ec - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp((fmin(Ev, EAccept) / KbT))));
	} else if (t_4 <= -2e-89) {
		tmp = t_3;
	} else if (t_4 <= -5e-221) {
		tmp = t_1 + (NaChar / (1.0 + Math.exp((fmax(Ev, EAccept) / KbT))));
	} else if (t_4 <= 1e-296) {
		tmp = NaChar / (Math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = fmin(Ev, EAccept) + Vef
	t_1 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_2 = NaChar / (1.0 + math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT)))
	t_3 = (NdChar / (1.0 + math.exp((mu / KbT)))) + t_2
	t_4 = t_1 + t_2
	tmp = 0
	if t_4 <= -2e+162:
		tmp = (NdChar / (1.0 + math.exp((-((Ec - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp((fmin(Ev, EAccept) / KbT))))
	elif t_4 <= -2e-89:
		tmp = t_3
	elif t_4 <= -5e-221:
		tmp = t_1 + (NaChar / (1.0 + math.exp((fmax(Ev, EAccept) / KbT))))
	elif t_4 <= 1e-296:
		tmp = NaChar / (math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0)
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(fmin(Ev, EAccept) + Vef)
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_2 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(t_0 + fmax(Ev, EAccept)) + Float64(-mu)) / KbT))))
	t_3 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + t_2)
	t_4 = Float64(t_1 + t_2)
	tmp = 0.0
	if (t_4 <= -2e+162)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Ec - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(fmin(Ev, EAccept) / KbT)))));
	elseif (t_4 <= -2e-89)
		tmp = t_3;
	elseif (t_4 <= -5e-221)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(fmax(Ev, EAccept) / KbT)))));
	elseif (t_4 <= 1e-296)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = min(Ev, EAccept) + Vef;
	t_1 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_2 = NaChar / (1.0 + exp((((t_0 + max(Ev, EAccept)) + -mu) / KbT)));
	t_3 = (NdChar / (1.0 + exp((mu / KbT)))) + t_2;
	t_4 = t_1 + t_2;
	tmp = 0.0;
	if (t_4 <= -2e+162)
		tmp = (NdChar / (1.0 + exp((-((Ec - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((min(Ev, EAccept) / KbT))));
	elseif (t_4 <= -2e-89)
		tmp = t_3;
	elseif (t_4 <= -5e-221)
		tmp = t_1 + (NaChar / (1.0 + exp((max(Ev, EAccept) / KbT))));
	elseif (t_4 <= 1e-296)
		tmp = NaChar / (exp((((max(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Min[Ev, EAccept], $MachinePrecision] + Vef), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(t$95$0 + N[Max[Ev, EAccept], $MachinePrecision]), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+162], N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(Ec - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[Min[Ev, EAccept], $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -2e-89], t$95$3, If[LessEqual[t$95$4, -5e-221], N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(N[Max[Ev, EAccept], $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e-296], N[(NaChar / N[(N[Exp[N[(N[(N[(N[Max[Ev, EAccept], $MachinePrecision] + t$95$0), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.9999999999999999e162

   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. Taylor expanded in Ev around inf

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

      1. lower-/.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in Vef around 0

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

      1. lower--.f64 64.1

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

   7. Applied rewrites 64.1%

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

   if -1.9999999999999999e162 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.00000000000000008e-89 or 1e-296 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around inf

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

      1. lower-/.f64 69.4

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

   4. Applied rewrites 69.4%

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

   if -2.00000000000000008e-89 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.99999999999999996e-221

   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. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 69.2

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

   4. Applied rewrites 69.2%

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

   if -4.99999999999999996e-221 < (+.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))))) < 1e-296

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

Alternative 4: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ t_2 := t\_0 + t\_1\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-277}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-301}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + t\_1\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1
         (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))
        (t_2 (+ t_0 t_1)))
   (if (<= t_2 -4e-277)
     (+ t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
     (if (<= t_2 5e-301)
       (/ NaChar (- (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) -1.0))
       (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT)));
	double t_2 = t_0 + t_1;
	double tmp;
	if (t_2 <= -4e-277) {
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (t_2 <= 5e-301) {
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_1 = nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt)))
    t_2 = t_0 + t_1
    if (t_2 <= (-4d-277)) then
        tmp = t_0 + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (t_2 <= 5d-301) then
        tmp = nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) - (-1.0d0))
    else
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT)));
	double t_2 = t_0 + t_1;
	double tmp;
	if (t_2 <= -4e-277) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (t_2 <= 5e-301) {
		tmp = NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT)))
	t_2 = t_0 + t_1
	tmp = 0
	if t_2 <= -4e-277:
		tmp = t_0 + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif t_2 <= 5e-301:
		tmp = NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0)
	else:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT))))
	t_2 = Float64(t_0 + t_1)
	tmp = 0.0
	if (t_2 <= -4e-277)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (t_2 <= 5e-301)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT)));
	t_2 = t_0 + t_1;
	tmp = 0.0;
	if (t_2 <= -4e-277)
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (t_2 <= 5e-301)
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	else
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-277], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-301], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -3.99999999999999988e-277

   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. Taylor expanded in Ev around inf

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

      1. lower-/.f64 69.1

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

   4. Applied rewrites 69.1%

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

   if -3.99999999999999988e-277 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5.00000000000000013e-301

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 5.00000000000000013e-301 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 68.1

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

   4. Applied rewrites 68.1%

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

Alternative 5: 78.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(Ev, EAccept\right) + Vef\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_2 := t\_1 + \frac{NaChar}{1 + e^{\frac{\left(t\_0 + \mathsf{max}\left(Ev, EAccept\right)\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-277}:\\ \;\;\;\;t\_1 + \frac{NaChar}{1 + e^{\frac{\mathsf{min}\left(Ev, EAccept\right)}{KbT}}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\mathsf{max}\left(Ev, EAccept\right) + t\_0\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{NaChar}{1 + e^{\frac{\mathsf{max}\left(Ev, EAccept\right)}{KbT}}}\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (fmin Ev EAccept) Vef))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_2
         (+
          t_1
          (/
           NaChar
           (+ 1.0 (exp (/ (+ (+ t_0 (fmax Ev EAccept)) (- mu)) KbT)))))))
   (if (<= t_2 -4e-277)
     (+ t_1 (/ NaChar (+ 1.0 (exp (/ (fmin Ev EAccept) KbT)))))
     (if (<= t_2 5e-106)
       (/ NaChar (- (exp (/ (- (+ (fmax Ev EAccept) t_0) mu) KbT)) -1.0))
       (+ t_1 (/ NaChar (+ 1.0 (exp (/ (fmax Ev EAccept) KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_2 = t_1 + (NaChar / (1.0 + exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))));
	double tmp;
	if (t_2 <= -4e-277) {
		tmp = t_1 + (NaChar / (1.0 + exp((fmin(Ev, EAccept) / KbT))));
	} else if (t_2 <= 5e-106) {
		tmp = NaChar / (exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_1 + (NaChar / (1.0 + exp((fmax(Ev, EAccept) / KbT))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(ev, eaccept) + vef
    t_1 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_2 = t_1 + (nachar / (1.0d0 + exp((((t_0 + fmax(ev, eaccept)) + -mu) / kbt))))
    if (t_2 <= (-4d-277)) then
        tmp = t_1 + (nachar / (1.0d0 + exp((fmin(ev, eaccept) / kbt))))
    else if (t_2 <= 5d-106) then
        tmp = nachar / (exp((((fmax(ev, eaccept) + t_0) - mu) / kbt)) - (-1.0d0))
    else
        tmp = t_1 + (nachar / (1.0d0 + exp((fmax(ev, eaccept) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_2 = t_1 + (NaChar / (1.0 + Math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))));
	double tmp;
	if (t_2 <= -4e-277) {
		tmp = t_1 + (NaChar / (1.0 + Math.exp((fmin(Ev, EAccept) / KbT))));
	} else if (t_2 <= 5e-106) {
		tmp = NaChar / (Math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_1 + (NaChar / (1.0 + Math.exp((fmax(Ev, EAccept) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = fmin(Ev, EAccept) + Vef
	t_1 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_2 = t_1 + (NaChar / (1.0 + math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))))
	tmp = 0
	if t_2 <= -4e-277:
		tmp = t_1 + (NaChar / (1.0 + math.exp((fmin(Ev, EAccept) / KbT))))
	elif t_2 <= 5e-106:
		tmp = NaChar / (math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0)
	else:
		tmp = t_1 + (NaChar / (1.0 + math.exp((fmax(Ev, EAccept) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(fmin(Ev, EAccept) + Vef)
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_2 = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(t_0 + fmax(Ev, EAccept)) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_2 <= -4e-277)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(fmin(Ev, EAccept) / KbT)))));
	elseif (t_2 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(fmax(Ev, EAccept) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = min(Ev, EAccept) + Vef;
	t_1 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_2 = t_1 + (NaChar / (1.0 + exp((((t_0 + max(Ev, EAccept)) + -mu) / KbT))));
	tmp = 0.0;
	if (t_2 <= -4e-277)
		tmp = t_1 + (NaChar / (1.0 + exp((min(Ev, EAccept) / KbT))));
	elseif (t_2 <= 5e-106)
		tmp = NaChar / (exp((((max(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	else
		tmp = t_1 + (NaChar / (1.0 + exp((max(Ev, EAccept) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Min[Ev, EAccept], $MachinePrecision] + Vef), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(t$95$0 + N[Max[Ev, EAccept], $MachinePrecision]), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-277], N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(N[Min[Ev, EAccept], $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(N[Max[Ev, EAccept], $MachinePrecision] + t$95$0), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(N[Max[Ev, EAccept], $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -3.99999999999999988e-277

   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. Taylor expanded in Ev around inf

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

      1. lower-/.f64 69.1

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

   4. Applied rewrites 69.1%

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

   if -3.99999999999999988e-277 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 69.2

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

   4. Applied rewrites 69.2%

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

Alternative 6: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(Ev, EAccept\right) + Vef\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_2 := t\_1 + \frac{NaChar}{1 + e^{\frac{\mathsf{max}\left(Ev, EAccept\right)}{KbT}}}\\ t_3 := t\_1 + \frac{NaChar}{1 + e^{\frac{\left(t\_0 + \mathsf{max}\left(Ev, EAccept\right)\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-221}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\mathsf{max}\left(Ev, EAccept\right) + t\_0\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (fmin Ev EAccept) Vef))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_2 (+ t_1 (/ NaChar (+ 1.0 (exp (/ (fmax Ev EAccept) KbT))))))
        (t_3
         (+
          t_1
          (/
           NaChar
           (+ 1.0 (exp (/ (+ (+ t_0 (fmax Ev EAccept)) (- mu)) KbT)))))))
   (if (<= t_3 -5e-221)
     t_2
     (if (<= t_3 5e-106)
       (/ NaChar (- (exp (/ (- (+ (fmax Ev EAccept) t_0) mu) KbT)) -1.0))
       t_2))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_2 = t_1 + (NaChar / (1.0 + exp((fmax(Ev, EAccept) / KbT))));
	double t_3 = t_1 + (NaChar / (1.0 + exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))));
	double tmp;
	if (t_3 <= -5e-221) {
		tmp = t_2;
	} else if (t_3 <= 5e-106) {
		tmp = NaChar / (exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    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 = fmin(ev, eaccept) + vef
    t_1 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_2 = t_1 + (nachar / (1.0d0 + exp((fmax(ev, eaccept) / kbt))))
    t_3 = t_1 + (nachar / (1.0d0 + exp((((t_0 + fmax(ev, eaccept)) + -mu) / kbt))))
    if (t_3 <= (-5d-221)) then
        tmp = t_2
    else if (t_3 <= 5d-106) then
        tmp = nachar / (exp((((fmax(ev, eaccept) + t_0) - mu) / kbt)) - (-1.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_2 = t_1 + (NaChar / (1.0 + Math.exp((fmax(Ev, EAccept) / KbT))));
	double t_3 = t_1 + (NaChar / (1.0 + Math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))));
	double tmp;
	if (t_3 <= -5e-221) {
		tmp = t_2;
	} else if (t_3 <= 5e-106) {
		tmp = NaChar / (Math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = fmin(Ev, EAccept) + Vef
	t_1 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_2 = t_1 + (NaChar / (1.0 + math.exp((fmax(Ev, EAccept) / KbT))))
	t_3 = t_1 + (NaChar / (1.0 + math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))))
	tmp = 0
	if t_3 <= -5e-221:
		tmp = t_2
	elif t_3 <= 5e-106:
		tmp = NaChar / (math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0)
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(fmin(Ev, EAccept) + Vef)
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_2 = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(fmax(Ev, EAccept) / KbT)))))
	t_3 = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(t_0 + fmax(Ev, EAccept)) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_3 <= -5e-221)
		tmp = t_2;
	elseif (t_3 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = min(Ev, EAccept) + Vef;
	t_1 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_2 = t_1 + (NaChar / (1.0 + exp((max(Ev, EAccept) / KbT))));
	t_3 = t_1 + (NaChar / (1.0 + exp((((t_0 + max(Ev, EAccept)) + -mu) / KbT))));
	tmp = 0.0;
	if (t_3 <= -5e-221)
		tmp = t_2;
	elseif (t_3 <= 5e-106)
		tmp = NaChar / (exp((((max(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Min[Ev, EAccept], $MachinePrecision] + Vef), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(N[Max[Ev, EAccept], $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(t$95$0 + N[Max[Ev, EAccept], $MachinePrecision]), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-221], t$95$2, If[LessEqual[t$95$3, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(N[Max[Ev, EAccept], $MachinePrecision] + t$95$0), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.99999999999999996e-221 or 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 69.2

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

   4. Applied rewrites 69.2%

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

   if -4.99999999999999996e-221 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

Alternative 7: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(Ev, EAccept\right) + Vef\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(t\_0 + \mathsf{max}\left(Ev, EAccept\right)\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-277}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(Ec - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\mathsf{min}\left(Ev, EAccept\right)}{KbT}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\mathsf{max}\left(Ev, EAccept\right) + t\_0\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{t\_0 - mu}{KbT}}}\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (fmin Ev EAccept) Vef))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/
           NaChar
           (+ 1.0 (exp (/ (+ (+ t_0 (fmax Ev EAccept)) (- mu)) KbT)))))))
   (if (<= t_1 -4e-277)
     (+
      (/ NdChar (+ 1.0 (exp (/ (- (- (- Ec EDonor) mu)) KbT))))
      (/ NaChar (+ 1.0 (exp (/ (fmin Ev EAccept) KbT)))))
     (if (<= t_1 5e-106)
       (/ NaChar (- (exp (/ (- (+ (fmax Ev EAccept) t_0) mu) KbT)) -1.0))
       (+
        (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
        (/ NaChar (+ 1.0 (exp (/ (- t_0 mu) KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -4e-277) {
		tmp = (NdChar / (1.0 + exp((-((Ec - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((fmin(Ev, EAccept) / KbT))));
	} else if (t_1 <= 5e-106) {
		tmp = NaChar / (exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp(((t_0 - mu) / KbT))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    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 = fmin(ev, eaccept) + vef
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp((((t_0 + fmax(ev, eaccept)) + -mu) / kbt))))
    if (t_1 <= (-4d-277)) then
        tmp = (ndchar / (1.0d0 + exp((-((ec - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp((fmin(ev, eaccept) / kbt))))
    else if (t_1 <= 5d-106) then
        tmp = nachar / (exp((((fmax(ev, eaccept) + t_0) - mu) / kbt)) - (-1.0d0))
    else
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / (1.0d0 + exp(((t_0 - mu) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -4e-277) {
		tmp = (NdChar / (1.0 + Math.exp((-((Ec - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp((fmin(Ev, EAccept) / KbT))));
	} else if (t_1 <= 5e-106) {
		tmp = NaChar / (Math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / (1.0 + Math.exp(((t_0 - mu) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = fmin(Ev, EAccept) + Vef
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))))
	tmp = 0
	if t_1 <= -4e-277:
		tmp = (NdChar / (1.0 + math.exp((-((Ec - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp((fmin(Ev, EAccept) / KbT))))
	elif t_1 <= 5e-106:
		tmp = NaChar / (math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0)
	else:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / (1.0 + math.exp(((t_0 - mu) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(fmin(Ev, EAccept) + Vef)
	t_1 = 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(t_0 + fmax(Ev, EAccept)) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_1 <= -4e-277)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Ec - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(fmin(Ev, EAccept) / KbT)))));
	elseif (t_1 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(t_0 - mu) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = min(Ev, EAccept) + Vef;
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((((t_0 + max(Ev, EAccept)) + -mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -4e-277)
		tmp = (NdChar / (1.0 + exp((-((Ec - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((min(Ev, EAccept) / KbT))));
	elseif (t_1 <= 5e-106)
		tmp = NaChar / (exp((((max(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	else
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp(((t_0 - mu) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Min[Ev, EAccept], $MachinePrecision] + Vef), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(t$95$0 + N[Max[Ev, EAccept], $MachinePrecision]), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-277], N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(Ec - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[Min[Ev, EAccept], $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(N[Max[Ev, EAccept], $MachinePrecision] + t$95$0), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(t$95$0 - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -3.99999999999999988e-277

   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. Taylor expanded in Ev around inf

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

      1. lower-/.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in Vef around 0

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

      1. lower--.f64 64.1

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

   7. Applied rewrites 64.1%

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

   if -3.99999999999999988e-277 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around inf

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

      1. lower-/.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 69.7

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in EAccept around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 64.8

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

   10. Applied rewrites 64.8%

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

Alternative 8: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{-1 \cdot \frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -2.5 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq -7.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} - -1}\\ \mathbf{elif}\;mu \leq 6.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (* -1.0 (/ mu KbT))))))))
   (if (<= mu -2.5e+123)
     t_0
     (if (<= mu -7.5e+56)
       (/ NaChar (- (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) -1.0))
       (if (<= mu 6.5e+111)
         (+
          (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ 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 / KbT)))) + (NaChar / (1.0 + exp((-1.0 * (mu / KbT)))));
	double tmp;
	if (mu <= -2.5e+123) {
		tmp = t_0;
	} else if (mu <= -7.5e+56) {
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else if (mu <= 6.5e+111) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    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 / kbt)))) + (nachar / (1.0d0 + exp(((-1.0d0) * (mu / kbt)))))
    if (mu <= (-2.5d+123)) then
        tmp = t_0
    else if (mu <= (-7.5d+56)) then
        tmp = nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) - (-1.0d0))
    else if (mu <= 6.5d+111) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / (1.0d0 + exp((((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 / KbT)))) + (NaChar / (1.0 + Math.exp((-1.0 * (mu / KbT)))));
	double tmp;
	if (mu <= -2.5e+123) {
		tmp = t_0;
	} else if (mu <= -7.5e+56) {
		tmp = NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else if (mu <= 6.5e+111) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / (1.0 + Math.exp((((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 / KbT)))) + (NaChar / (1.0 + math.exp((-1.0 * (mu / KbT)))))
	tmp = 0
	if mu <= -2.5e+123:
		tmp = t_0
	elif mu <= -7.5e+56:
		tmp = NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0)
	elif mu <= 6.5e+111:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / (1.0 + math.exp((((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(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(-1.0 * Float64(mu / KbT))))))
	tmp = 0.0
	if (mu <= -2.5e+123)
		tmp = t_0;
	elseif (mu <= -7.5e+56)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) - -1.0));
	elseif (mu <= 6.5e+111)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(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 / KbT)))) + (NaChar / (1.0 + exp((-1.0 * (mu / KbT)))));
	tmp = 0.0;
	if (mu <= -2.5e+123)
		tmp = t_0;
	elseif (mu <= -7.5e+56)
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	elseif (mu <= 6.5e+111)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((((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[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(-1.0 * N[(mu / KbT), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -2.5e+123], t$95$0, If[LessEqual[mu, -7.5e+56], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 6.5e+111], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Ev + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -2.49999999999999987e123 or 6.5000000000000002e111 < mu

   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. Taylor expanded in mu around inf

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

      1. lower-/.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in mu around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -2.49999999999999987e123 < mu < -7.4999999999999999e56

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if -7.4999999999999999e56 < mu < 6.5000000000000002e111

   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. Taylor expanded in mu around inf

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

      1. lower-/.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 69.7

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in EAccept around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 64.8

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

   10. Applied rewrites 64.8%

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

Alternative 9: 69.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(Ec - Vef\right) - EDonor\\ t_1 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{-\left(t\_0 - mu\right)}{KbT}}} + t\_1\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NaChar, \frac{NdChar}{e^{\frac{mu - t\_0}{KbT}} - -1}\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-277}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{-1 \cdot \frac{mu}{KbT}}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + t\_1\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (- Ec Vef) EDonor))
        (t_1
         (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))
        (t_2 (+ (/ NdChar (+ 1.0 (exp (/ (- (- t_0 mu)) KbT)))) t_1)))
   (if (<= t_2 -2e+162)
     (fma 0.5 NaChar (/ NdChar (- (exp (/ (- mu t_0) KbT)) -1.0)))
     (if (<= t_2 -4e-277)
       (+
        (/ NdChar (+ 1.0 (exp (/ mu KbT))))
        (/ NaChar (+ 1.0 (exp (* -1.0 (/ mu KbT))))))
       (if (<= t_2 5e-106)
         (/ NaChar (- (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) -1.0))
         (+ (* 0.5 NdChar) 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 = (Ec - Vef) - EDonor;
	double t_1 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT)));
	double t_2 = (NdChar / (1.0 + exp((-(t_0 - mu) / KbT)))) + t_1;
	double tmp;
	if (t_2 <= -2e+162) {
		tmp = fma(0.5, NaChar, (NdChar / (exp(((mu - t_0) / KbT)) - -1.0)));
	} else if (t_2 <= -4e-277) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-1.0 * (mu / KbT)))));
	} else if (t_2 <= 5e-106) {
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Ec - Vef) - EDonor)
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT))))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(t_0 - mu)) / KbT)))) + t_1)
	tmp = 0.0
	if (t_2 <= -2e+162)
		tmp = fma(0.5, NaChar, Float64(NdChar / Float64(exp(Float64(Float64(mu - t_0) / KbT)) - -1.0)));
	elseif (t_2 <= -4e-277)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(-1.0 * Float64(mu / KbT))))));
	elseif (t_2 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + t_1);
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(t$95$0 - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+162], N[(0.5 * NaChar + N[(NdChar / N[(N[Exp[N[(N[(mu - t$95$0), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-277], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(-1.0 * N[(mu / KbT), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.9999999999999999e162

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.3%

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

   if -1.9999999999999999e162 < (+.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))))) < -3.99999999999999988e-277

   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. Taylor expanded in mu around inf

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

      1. lower-/.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in mu around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -3.99999999999999988e-277 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

Alternative 10: 67.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(Ev, EAccept\right) + Vef\\ t_1 := \left(Ec - Vef\right) - EDonor\\ t_2 := \frac{NaChar}{1 + e^{\frac{\left(t\_0 + \mathsf{max}\left(Ev, EAccept\right)\right) + \left(-mu\right)}{KbT}}}\\ t_3 := \frac{NdChar}{1 + e^{\frac{-\left(t\_1 - mu\right)}{KbT}}} + t\_2\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NaChar, \frac{NdChar}{e^{\frac{mu - t\_1}{KbT}} - -1}\right)\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\mathsf{max}\left(Ev, EAccept\right)}{KbT}}}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\mathsf{max}\left(Ev, EAccept\right) + t\_0\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + t\_2\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (fmin Ev EAccept) Vef))
        (t_1 (- (- Ec Vef) EDonor))
        (t_2
         (/ NaChar (+ 1.0 (exp (/ (+ (+ t_0 (fmax Ev EAccept)) (- mu)) KbT)))))
        (t_3 (+ (/ NdChar (+ 1.0 (exp (/ (- (- t_1 mu)) KbT)))) t_2)))
   (if (<= t_3 -2e+162)
     (fma 0.5 NaChar (/ NdChar (- (exp (/ (- mu t_1) KbT)) -1.0)))
     (if (<= t_3 -4e+17)
       (+
        (/ NdChar (+ 1.0 (exp (/ mu KbT))))
        (/ NaChar (+ 1.0 (exp (/ (fmax Ev EAccept) KbT)))))
       (if (<= t_3 5e-106)
         (/ NaChar (- (exp (/ (- (+ (fmax Ev EAccept) t_0) mu) KbT)) -1.0))
         (+ (* 0.5 NdChar) t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = (Ec - Vef) - EDonor;
	double t_2 = NaChar / (1.0 + exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT)));
	double t_3 = (NdChar / (1.0 + exp((-(t_1 - mu) / KbT)))) + t_2;
	double tmp;
	if (t_3 <= -2e+162) {
		tmp = fma(0.5, NaChar, (NdChar / (exp(((mu - t_1) / KbT)) - -1.0)));
	} else if (t_3 <= -4e+17) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((fmax(Ev, EAccept) / KbT))));
	} else if (t_3 <= 5e-106) {
		tmp = NaChar / (exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + t_2;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(fmin(Ev, EAccept) + Vef)
	t_1 = Float64(Float64(Ec - Vef) - EDonor)
	t_2 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(t_0 + fmax(Ev, EAccept)) + Float64(-mu)) / KbT))))
	t_3 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(t_1 - mu)) / KbT)))) + t_2)
	tmp = 0.0
	if (t_3 <= -2e+162)
		tmp = fma(0.5, NaChar, Float64(NdChar / Float64(exp(Float64(Float64(mu - t_1) / KbT)) - -1.0)));
	elseif (t_3 <= -4e+17)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(fmax(Ev, EAccept) / KbT)))));
	elseif (t_3 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + t_2);
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Min[Ev, EAccept], $MachinePrecision] + Vef), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(t$95$0 + N[Max[Ev, EAccept], $MachinePrecision]), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(t$95$1 - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+162], N[(0.5 * NaChar + N[(NdChar / N[(N[Exp[N[(N[(mu - t$95$1), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -4e+17], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[Max[Ev, EAccept], $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(N[Max[Ev, EAccept], $MachinePrecision] + t$95$0), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.9999999999999999e162

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.3%

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

   if -1.9999999999999999e162 < (+.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))))) < -4e17

   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. Taylor expanded in mu around inf

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

      1. lower-/.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 49.1

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

   7. Applied rewrites 49.1%

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

   if -4e17 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

Alternative 11: 67.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(Ec - Vef\right) - EDonor\\ t_1 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{-\left(t\_0 - mu\right)}{KbT}}} + t\_1\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NaChar, \frac{NdChar}{e^{\frac{mu - t\_0}{KbT}} - -1}\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + t\_1\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (- Ec Vef) EDonor))
        (t_1
         (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))
        (t_2 (+ (/ NdChar (+ 1.0 (exp (/ (- (- t_0 mu)) KbT)))) t_1)))
   (if (<= t_2 -4e+17)
     (fma 0.5 NaChar (/ NdChar (- (exp (/ (- mu t_0) KbT)) -1.0)))
     (if (<= t_2 5e-106)
       (/ NaChar (- (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) -1.0))
       (+ (* 0.5 NdChar) 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 = (Ec - Vef) - EDonor;
	double t_1 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT)));
	double t_2 = (NdChar / (1.0 + exp((-(t_0 - mu) / KbT)))) + t_1;
	double tmp;
	if (t_2 <= -4e+17) {
		tmp = fma(0.5, NaChar, (NdChar / (exp(((mu - t_0) / KbT)) - -1.0)));
	} else if (t_2 <= 5e-106) {
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Ec - Vef) - EDonor)
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT))))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(t_0 - mu)) / KbT)))) + t_1)
	tmp = 0.0
	if (t_2 <= -4e+17)
		tmp = fma(0.5, NaChar, Float64(NdChar / Float64(exp(Float64(Float64(mu - t_0) / KbT)) - -1.0)));
	elseif (t_2 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + t_1);
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(t$95$0 - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+17], N[(0.5 * NaChar + N[(NdChar / N[(N[Exp[N[(N[(mu - t$95$0), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4e17

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.3%

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

   if -4e17 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

Alternative 12: 66.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(Ec - Vef\right) - EDonor\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(t\_0 - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NaChar, \frac{NdChar}{e^{\frac{mu - t\_0}{KbT}} - -1}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Vef\right) - mu}{KbT}}}\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (- Ec Vef) EDonor))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- t_0 mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -4e+17)
     (fma 0.5 NaChar (/ NdChar (- (exp (/ (- mu t_0) KbT)) -1.0)))
     (if (<= t_1 5e-106)
       (/ NaChar (- (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) -1.0))
       (+
        (* 0.5 NdChar)
        (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept Vef) mu) KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Ec - Vef) - EDonor;
	double t_1 = (NdChar / (1.0 + exp((-(t_0 - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -4e+17) {
		tmp = fma(0.5, NaChar, (NdChar / (exp(((mu - t_0) / KbT)) - -1.0)));
	} else if (t_1 <= 5e-106) {
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((((EAccept + Vef) - mu) / KbT))));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Ec - Vef) - EDonor)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(t_0 - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_1 <= -4e+17)
		tmp = fma(0.5, NaChar, Float64(NdChar / Float64(exp(Float64(Float64(mu - t_0) / KbT)) - -1.0)));
	elseif (t_1 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Vef) - mu) / KbT)))));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(t$95$0 - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+17], N[(0.5 * NaChar + N[(NdChar / N[(N[Exp[N[(N[(mu - t$95$0), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4e17

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.3%

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

   if -4e17 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 42.9

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

   7. Applied rewrites 42.9%

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

Alternative 13: 65.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Vef\right) - mu}{KbT}}}\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_0 -4e+17)
     (+ (* 0.5 NdChar) (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept Ev) mu) KbT)))))
     (if (<= t_0 5e-106)
       (/ NaChar (- (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) -1.0))
       (+
        (* 0.5 NdChar)
        (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept Vef) mu) KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_0 <= -4e+17) {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT))));
	} else if (t_0 <= 5e-106) {
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((((EAccept + Vef) - mu) / KbT))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (t_0 <= (-4d+17)) then
        tmp = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp((((eaccept + ev) - mu) / kbt))))
    else if (t_0 <= 5d-106) then
        tmp = nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) - (-1.0d0))
    else
        tmp = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp((((eaccept + vef) - mu) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_0 <= -4e+17) {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp((((EAccept + Ev) - mu) / KbT))));
	} else if (t_0 <= 5e-106) {
		tmp = NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp((((EAccept + Vef) - mu) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))
	tmp = 0
	if t_0 <= -4e+17:
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + math.exp((((EAccept + Ev) - mu) / KbT))))
	elif t_0 <= 5e-106:
		tmp = NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0)
	else:
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + math.exp((((EAccept + Vef) - mu) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_0 <= -4e+17)
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)))));
	elseif (t_0 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Vef) - mu) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	tmp = 0.0;
	if (t_0 <= -4e+17)
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT))));
	elseif (t_0 <= 5e-106)
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	else
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp((((EAccept + Vef) - mu) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+17], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + Vef), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4e17

   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. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in Vef around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 43.1

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

   7. Applied rewrites 43.1%

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

   if -4e17 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 42.9

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

   7. Applied rewrites 42.9%

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

Alternative 14: 65.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (* 0.5 NdChar)
          (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept Ev) mu) KbT))))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -4e+17)
     t_0
     (if (<= t_1 5e-106)
       (/ NaChar (- (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) -1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (0.5 * NdChar) + (NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT))));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -4e+17) {
		tmp = t_0;
	} else if (t_1 <= 5e-106) {
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp((((eaccept + ev) - mu) / kbt))))
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (t_1 <= (-4d+17)) then
        tmp = t_0
    else if (t_1 <= 5d-106) then
        tmp = nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp((((EAccept + Ev) - mu) / KbT))));
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -4e+17) {
		tmp = t_0;
	} else if (t_1 <= 5e-106) {
		tmp = NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (0.5 * NdChar) + (NaChar / (1.0 + math.exp((((EAccept + Ev) - mu) / KbT))))
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))
	tmp = 0
	if t_1 <= -4e+17:
		tmp = t_0
	elif t_1 <= 5e-106:
		tmp = NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_1 <= -4e+17)
		tmp = t_0;
	elseif (t_1 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (0.5 * NdChar) + (NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT))));
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -4e+17)
		tmp = t_0;
	elseif (t_1 <= 5e-106)
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+17], t$95$0, If[LessEqual[t$95$1, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $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))))) < -4e17 or 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in Vef around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 43.1

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

   7. Applied rewrites 43.1%

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

   if -4e17 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

Alternative 15: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(Ev, EAccept\right) + Vef\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(t\_0 + \mathsf{max}\left(Ev, EAccept\right)\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{\mathsf{max}\left(Ev, EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\mathsf{max}\left(Ev, EAccept\right) + t\_0\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{Vef - mu}{KbT}}}\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (fmin Ev EAccept) Vef))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/
           NaChar
           (+ 1.0 (exp (/ (+ (+ t_0 (fmax Ev EAccept)) (- mu)) KbT)))))))
   (if (<= t_1 -4e+17)
     (+
      (* 0.5 NdChar)
      (/ NaChar (+ 1.0 (exp (/ (- (fmax Ev EAccept) mu) KbT)))))
     (if (<= t_1 5e-106)
       (/ NaChar (- (exp (/ (- (+ (fmax Ev EAccept) t_0) mu) KbT)) -1.0))
       (+ (* 0.5 NdChar) (/ NaChar (+ 1.0 (exp (/ (- Vef mu) KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -4e+17) {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp(((fmax(Ev, EAccept) - mu) / KbT))));
	} else if (t_1 <= 5e-106) {
		tmp = NaChar / (exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp(((Vef - mu) / KbT))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    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 = fmin(ev, eaccept) + vef
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp((((t_0 + fmax(ev, eaccept)) + -mu) / kbt))))
    if (t_1 <= (-4d+17)) then
        tmp = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp(((fmax(ev, eaccept) - mu) / kbt))))
    else if (t_1 <= 5d-106) then
        tmp = nachar / (exp((((fmax(ev, eaccept) + t_0) - mu) / kbt)) - (-1.0d0))
    else
        tmp = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp(((vef - mu) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = fmin(Ev, EAccept) + Vef;
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -4e+17) {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp(((fmax(Ev, EAccept) - mu) / KbT))));
	} else if (t_1 <= 5e-106) {
		tmp = NaChar / (Math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp(((Vef - mu) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = fmin(Ev, EAccept) + Vef
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp((((t_0 + fmax(Ev, EAccept)) + -mu) / KbT))))
	tmp = 0
	if t_1 <= -4e+17:
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + math.exp(((fmax(Ev, EAccept) - mu) / KbT))))
	elif t_1 <= 5e-106:
		tmp = NaChar / (math.exp((((fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0)
	else:
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + math.exp(((Vef - mu) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(fmin(Ev, EAccept) + Vef)
	t_1 = 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(t_0 + fmax(Ev, EAccept)) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_1 <= -4e+17)
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(fmax(Ev, EAccept) - mu) / KbT)))));
	elseif (t_1 <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(fmax(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef - mu) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = min(Ev, EAccept) + Vef;
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((((t_0 + max(Ev, EAccept)) + -mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -4e+17)
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp(((max(Ev, EAccept) - mu) / KbT))));
	elseif (t_1 <= 5e-106)
		tmp = NaChar / (exp((((max(Ev, EAccept) + t_0) - mu) / KbT)) - -1.0);
	else
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp(((Vef - mu) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Min[Ev, EAccept], $MachinePrecision] + Vef), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(t$95$0 + N[Max[Ev, EAccept], $MachinePrecision]), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+17], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[Max[Ev, EAccept], $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(N[Max[Ev, EAccept], $MachinePrecision] + t$95$0), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4e17

   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. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 42.9

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

   7. Applied rewrites 42.9%

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

   8. Taylor expanded in Vef around 0

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

   10. Applied rewrites 39.5%

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

   if -4e17 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 42.9

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

   7. Applied rewrites 42.9%

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

   8. Taylor expanded in EAccept around 0

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

      1. lower--.f64 39.0

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

   10. Applied rewrites 39.0%

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

Alternative 16: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{Vef - mu}{KbT}}}\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<=
      (+
       (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
       (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))
      5e-106)
   (/ NaChar (- (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)) -1.0))
   (+ (* 0.5 NdChar) (/ NaChar (+ 1.0 (exp (/ (- Vef mu) KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (((NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))) <= 5e-106) {
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp(((Vef - mu) / KbT))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    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 / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))) <= 5d-106) then
        tmp = nachar / (exp((((eaccept + (ev + vef)) - mu) / kbt)) - (-1.0d0))
    else
        tmp = (0.5d0 * ndchar) + (nachar / (1.0d0 + exp(((vef - mu) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (((NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))) <= 5e-106) {
		tmp = NaChar / (Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + Math.exp(((Vef - mu) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if ((NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))) <= 5e-106:
		tmp = NaChar / (math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0)
	else:
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + math.exp(((Vef - mu) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (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))))) <= 5e-106)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef - mu) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (((NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))) <= 5e-106)
		tmp = NaChar / (exp((((EAccept + (Ev + Vef)) - mu) / KbT)) - -1.0);
	else
		tmp = (0.5 * NdChar) + (NaChar / (1.0 + exp(((Vef - mu) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[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], 5e-106], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999983e-106

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 4.99999999999999983e-106 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 42.9

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

   7. Applied rewrites 42.9%

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

   8. Taylor expanded in EAccept around 0

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

      1. lower--.f64 39.0

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

   10. Applied rewrites 39.0%

       \[\leadsto 0.5 \cdot NdChar + \frac{NaChar}{1 + e^{\frac{Vef - mu}{KbT}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 60.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;KbT \leq 1.4 \cdot 10^{+206}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(\mathsf{max}\left(Ev, EAccept\right) + \left(\mathsf{min}\left(Ev, EAccept\right) + Vef\right)\right) - mu}{KbT}} - -1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{\left(2 + \left(\frac{\mathsf{max}\left(Ev, EAccept\right)}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT 1.4e+206)
   (/
    NaChar
    (-
     (exp (/ (- (+ (fmax Ev EAccept) (+ (fmin Ev EAccept) Vef)) mu) KbT))
     -1.0))
   (+
    (* 0.5 NdChar)
    (/
     NaChar
     (- (+ 2.0 (+ (/ (fmax Ev EAccept) KbT) (/ Vef KbT))) (/ mu KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= 1.4e+206) {
		tmp = NaChar / (exp((((fmax(Ev, EAccept) + (fmin(Ev, EAccept) + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / ((2.0 + ((fmax(Ev, EAccept) / KbT) + (Vef / KbT))) - (mu / KbT)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= 1.4d+206) then
        tmp = nachar / (exp((((fmax(ev, eaccept) + (fmin(ev, eaccept) + vef)) - mu) / kbt)) - (-1.0d0))
    else
        tmp = (0.5d0 * ndchar) + (nachar / ((2.0d0 + ((fmax(ev, eaccept) / kbt) + (vef / kbt))) - (mu / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= 1.4e+206) {
		tmp = NaChar / (Math.exp((((fmax(Ev, EAccept) + (fmin(Ev, EAccept) + Vef)) - mu) / KbT)) - -1.0);
	} else {
		tmp = (0.5 * NdChar) + (NaChar / ((2.0 + ((fmax(Ev, EAccept) / KbT) + (Vef / KbT))) - (mu / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= 1.4e+206:
		tmp = NaChar / (math.exp((((fmax(Ev, EAccept) + (fmin(Ev, EAccept) + Vef)) - mu) / KbT)) - -1.0)
	else:
		tmp = (0.5 * NdChar) + (NaChar / ((2.0 + ((fmax(Ev, EAccept) / KbT) + (Vef / KbT))) - (mu / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= 1.4e+206)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(fmax(Ev, EAccept) + Float64(fmin(Ev, EAccept) + Vef)) - mu) / KbT)) - -1.0));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(fmax(Ev, EAccept) / KbT) + Float64(Vef / KbT))) - Float64(mu / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= 1.4e+206)
		tmp = NaChar / (exp((((max(Ev, EAccept) + (min(Ev, EAccept) + Vef)) - mu) / KbT)) - -1.0);
	else
		tmp = (0.5 * NdChar) + (NaChar / ((2.0 + ((max(Ev, EAccept) / KbT) + (Vef / KbT))) - (mu / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, 1.4e+206], N[(NaChar / N[(N[Exp[N[(N[(N[(N[Max[Ev, EAccept], $MachinePrecision] + N[(N[Min[Ev, EAccept], $MachinePrecision] + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(N[(2.0 + N[(N[(N[Max[Ev, EAccept], $MachinePrecision] / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < 1.3999999999999999e206

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if 1.3999999999999999e206 < KbT

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 42.9

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

   7. Applied rewrites 42.9%

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

   8. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 25.7

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

   10. Applied rewrites 25.7%

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

Alternative 18: 53.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;KbT \leq 7.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(\mathsf{max}\left(Ev, EAccept\right) + \mathsf{min}\left(Ev, EAccept\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar + \frac{NaChar}{\left(2 + \left(\frac{\mathsf{max}\left(Ev, EAccept\right)}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT 7.8e+205)
   (/
    NaChar
    (+ 1.0 (exp (/ (- (+ (fmax Ev EAccept) (fmin Ev EAccept)) mu) KbT))))
   (+
    (* 0.5 NdChar)
    (/
     NaChar
     (- (+ 2.0 (+ (/ (fmax Ev EAccept) KbT) (/ Vef KbT))) (/ mu KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= 7.8e+205) {
		tmp = NaChar / (1.0 + exp((((fmax(Ev, EAccept) + fmin(Ev, EAccept)) - mu) / KbT)));
	} else {
		tmp = (0.5 * NdChar) + (NaChar / ((2.0 + ((fmax(Ev, EAccept) / KbT) + (Vef / KbT))) - (mu / KbT)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= 7.8d+205) then
        tmp = nachar / (1.0d0 + exp((((fmax(ev, eaccept) + fmin(ev, eaccept)) - mu) / kbt)))
    else
        tmp = (0.5d0 * ndchar) + (nachar / ((2.0d0 + ((fmax(ev, eaccept) / kbt) + (vef / kbt))) - (mu / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= 7.8e+205) {
		tmp = NaChar / (1.0 + Math.exp((((fmax(Ev, EAccept) + fmin(Ev, EAccept)) - mu) / KbT)));
	} else {
		tmp = (0.5 * NdChar) + (NaChar / ((2.0 + ((fmax(Ev, EAccept) / KbT) + (Vef / KbT))) - (mu / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= 7.8e+205:
		tmp = NaChar / (1.0 + math.exp((((fmax(Ev, EAccept) + fmin(Ev, EAccept)) - mu) / KbT)))
	else:
		tmp = (0.5 * NdChar) + (NaChar / ((2.0 + ((fmax(Ev, EAccept) / KbT) + (Vef / KbT))) - (mu / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= 7.8e+205)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(fmax(Ev, EAccept) + fmin(Ev, EAccept)) - mu) / KbT))));
	else
		tmp = Float64(Float64(0.5 * NdChar) + Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(fmax(Ev, EAccept) / KbT) + Float64(Vef / KbT))) - Float64(mu / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= 7.8e+205)
		tmp = NaChar / (1.0 + exp((((max(Ev, EAccept) + min(Ev, EAccept)) - mu) / KbT)));
	else
		tmp = (0.5 * NdChar) + (NaChar / ((2.0 + ((max(Ev, EAccept) / KbT) + (Vef / KbT))) - (mu / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, 7.8e+205], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[Max[Ev, EAccept], $MachinePrecision] + N[Min[Ev, EAccept], $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * NdChar), $MachinePrecision] + N[(NaChar / N[(N[(2.0 + N[(N[(N[Max[Ev, EAccept], $MachinePrecision] / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < 7.7999999999999997e205

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in Vef around 0

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

   7. Applied rewrites 51.3%

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

   if 7.7999999999999997e205 < KbT

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 42.9

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

   7. Applied rewrites 42.9%

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

   8. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 25.7

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

   10. Applied rewrites 25.7%

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

Alternative 19: 53.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;KbT \leq 6.4 \cdot 10^{+205}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\left(NdChar + NaChar\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT 6.4e+205)
   (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept Ev) mu) KbT))))
   (* (+ NdChar NaChar) 0.5)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= 6.4e+205) {
		tmp = NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT)));
	} else {
		tmp = (NdChar + NaChar) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= 6.4d+205) then
        tmp = nachar / (1.0d0 + exp((((eaccept + ev) - mu) / kbt)))
    else
        tmp = (ndchar + nachar) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= 6.4e+205) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + Ev) - mu) / KbT)));
	} else {
		tmp = (NdChar + NaChar) * 0.5;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= 6.4e+205:
		tmp = NaChar / (1.0 + math.exp((((EAccept + Ev) - mu) / KbT)))
	else:
		tmp = (NdChar + NaChar) * 0.5
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= 6.4e+205)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT))));
	else
		tmp = Float64(Float64(NdChar + NaChar) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= 6.4e+205)
		tmp = NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT)));
	else
		tmp = (NdChar + NaChar) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, 6.4e+205], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar + NaChar), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < 6.39999999999999993e205

   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. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in Vef around 0

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

   7. Applied rewrites 51.3%

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

   if 6.39999999999999993e205 < KbT

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.6

         \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 27.6

         \[\leadsto \left(NdChar + NaChar\right) \cdot 0.5 \]

   6. Applied rewrites 27.6%

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

Alternative 20: 32.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(NdChar + NaChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-264}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{\frac{2 + -2 \cdot \frac{NaChar}{NdChar}}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NdChar NaChar) 0.5))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -5e-264)
     t_0
     (if (<= t_1 5e-228)
       (/ 1.0 (/ (+ 2.0 (* -2.0 (/ NaChar NdChar))) 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 = (NdChar + NaChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -5e-264) {
		tmp = t_0;
	} else if (t_1 <= 5e-228) {
		tmp = 1.0 / ((2.0 + (-2.0 * (NaChar / NdChar))) / NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (ndchar + nachar) * 0.5d0
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (t_1 <= (-5d-264)) then
        tmp = t_0
    else if (t_1 <= 5d-228) then
        tmp = 1.0d0 / ((2.0d0 + ((-2.0d0) * (nachar / ndchar))) / ndchar)
    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 + NaChar) * 0.5;
	double t_1 = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -5e-264) {
		tmp = t_0;
	} else if (t_1 <= 5e-228) {
		tmp = 1.0 / ((2.0 + (-2.0 * (NaChar / NdChar))) / NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar + NaChar) * 0.5
	t_1 = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))
	tmp = 0
	if t_1 <= -5e-264:
		tmp = t_0
	elif t_1 <= 5e-228:
		tmp = 1.0 / ((2.0 + (-2.0 * (NaChar / NdChar))) / NdChar)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar + NaChar) * 0.5)
	t_1 = 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)))))
	tmp = 0.0
	if (t_1 <= -5e-264)
		tmp = t_0;
	elseif (t_1 <= 5e-228)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(-2.0 * Float64(NaChar / NdChar))) / NdChar));
	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 + NaChar) * 0.5;
	t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -5e-264)
		tmp = t_0;
	elseif (t_1 <= 5e-228)
		tmp = 1.0 / ((2.0 + (-2.0 * (NaChar / NdChar))) / NdChar);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar + NaChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-264], t$95$0, If[LessEqual[t$95$1, 5e-228], N[(1.0 / N[(N[(2.0 + N[(-2.0 * N[(NaChar / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / NdChar), $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))))) < -5.0000000000000001e-264 or 4.99999999999999972e-228 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.6

         \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 27.6

         \[\leadsto \left(NdChar + NaChar\right) \cdot 0.5 \]

   6. Applied rewrites 27.6%

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

   if -5.0000000000000001e-264 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999972e-228

   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. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.6

         \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-unsound--.f64 N/A

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

      11. lower-*.f64 17.2

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

   6. Applied rewrites 17.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 17.2

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out-- N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 17.2

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. swap-sqr N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. swap-sqr N/A

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

      21. distribute-lft-out-- N/A

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

   8. Applied rewrites 17.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(NdChar - NaChar\right) \cdot 0.5}{0.25 \cdot \left(NdChar \cdot NdChar - NaChar \cdot NaChar\right)}}} \]

   9. Taylor expanded in NdChar around inf

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 20.1

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

   11. Applied rewrites 20.1%

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

Alternative 21: 27.6% accurate, 9.1× speedup

Math

\[\left(NdChar + NaChar\right) \cdot 0.5 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    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 + nachar) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in KbT around inf

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 27.6

      \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

4. Applied rewrites 27.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 27.6

      \[\leadsto \left(NdChar + NaChar\right) \cdot 0.5 \]

6. Applied rewrites 27.6%

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

Alternative 22: 18.1% accurate, 15.4× speedup

Math

\[0.5 \cdot NaChar \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in KbT around inf

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 27.6

      \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

4. Applied rewrites 27.6%

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

5. Taylor expanded in NdChar around 0

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

   1. lower-*.f64 18.1

      \[\leadsto 0.5 \cdot NaChar \]

7. Applied rewrites 18.1%

   \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(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))))))