Bulmash initializePoisson

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 23

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 9.535346565989641e-76
  2. Ec = -7.049381688849528e+183
  3. Vef = -1.38054472890095e-58
  4. EDonor = -2.1340783791844105e-111
  5. mu = -3.518034764667867e-23
  6. KbT = -1.3161378169435097e+22
  7. NaChar = 2.464608943697297e-76
  8. Ev = 2.580862885945627e+285
  9. EAccept = 6.271861122781451e-37

Specification

Math

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

FPCore

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

C

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

Fortran

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{Vef + mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}}\\ \mathbf{if}\;mu \leq -2.45 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq 1.05 \cdot 10^{+225}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + Vef\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ Vef mu) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ Ev Vef) mu) KbT)))))))
   (if (<= mu -2.45e+61)
     t_0
     (if (<= mu 1.05e+225)
       (+
        (/ NaChar (+ 1.0 (exp (/ (+ EAccept (+ Ev Vef)) KbT))))
        (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) KbT)))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	double tmp;
	if (mu <= -2.45e+61) {
		tmp = t_0;
	} else if (mu <= 1.05e+225) {
		tmp = (NaChar / (1.0 + exp(((EAccept + (Ev + Vef)) / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / 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(((vef + mu) / kbt)))) + (nachar / (1.0d0 + exp((((ev + vef) - mu) / kbt))))
    if (mu <= (-2.45d+61)) then
        tmp = t_0
    else if (mu <= 1.05d+225) then
        tmp = (nachar / (1.0d0 + exp(((eaccept + (ev + vef)) / kbt)))) + (ndchar / (1.0d0 + exp((((edonor + vef) - ec) / kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + Math.exp((((Ev + Vef) - mu) / KbT))));
	double tmp;
	if (mu <= -2.45e+61) {
		tmp = t_0;
	} else if (mu <= 1.05e+225) {
		tmp = (NaChar / (1.0 + Math.exp(((EAccept + (Ev + Vef)) / KbT)))) + (NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / 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(((Vef + mu) / KbT)))) + (NaChar / (1.0 + math.exp((((Ev + Vef) - mu) / KbT))))
	tmp = 0
	if mu <= -2.45e+61:
		tmp = t_0
	elif mu <= 1.05e+225:
		tmp = (NaChar / (1.0 + math.exp(((EAccept + (Ev + Vef)) / KbT)))) + (NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / 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(Float64(Vef + mu) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Ev + Vef) - mu) / KbT)))))
	tmp = 0.0
	if (mu <= -2.45e+61)
		tmp = t_0;
	elseif (mu <= 1.05e+225)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Float64(Ev + Vef)) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	tmp = 0.0;
	if (mu <= -2.45e+61)
		tmp = t_0;
	elseif (mu <= 1.05e+225)
		tmp = (NaChar / (1.0 + exp(((EAccept + (Ev + Vef)) / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(Vef + mu), $MachinePrecision] / 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]}, If[LessEqual[mu, -2.45e+61], t$95$0, If[LessEqual[mu, 1.05e+225], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if mu < -2.45000000000000013e61 or 1.05e225 < mu

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in EDonor around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 43.3

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in Ec around 0

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

      1. lift-+.f64 39.3

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

   10. Applied rewrites 39.3%

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

   11. Taylor expanded in EAccept around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lift-+.f64 76.3

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

   13. Applied rewrites 76.3%

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

   if -2.45000000000000013e61 < mu < 1.05e225

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

Alternative 3: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;EDonor \leq -1.52 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;EDonor \leq -8.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ \mathbf{elif}\;EDonor \leq 7.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef + mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= EDonor -1.52e+159)
     t_0
     (if (<= EDonor -8.2e+73)
       (+
        (/ NaChar (+ 1.0 (exp (/ Vef KbT))))
        (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) KbT)))))
       (if (<= EDonor 7.2e+102)
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ Vef mu) 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((EDonor / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (EDonor <= -1.52e+159) {
		tmp = t_0;
	} else if (EDonor <= -8.2e+73) {
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT))));
	} else if (EDonor <= 7.2e+102) {
		tmp = (NdChar / (1.0 + exp(((Vef + mu) / 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((edonor / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (edonor <= (-1.52d+159)) then
        tmp = t_0
    else if (edonor <= (-8.2d+73)) then
        tmp = (nachar / (1.0d0 + exp((vef / kbt)))) + (ndchar / (1.0d0 + exp((((edonor + vef) - ec) / kbt))))
    else if (edonor <= 7.2d+102) then
        tmp = (ndchar / (1.0d0 + exp(((vef + mu) / 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((EDonor / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (EDonor <= -1.52e+159) {
		tmp = t_0;
	} else if (EDonor <= -8.2e+73) {
		tmp = (NaChar / (1.0 + Math.exp((Vef / KbT)))) + (NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / KbT))));
	} else if (EDonor <= 7.2e+102) {
		tmp = (NdChar / (1.0 + Math.exp(((Vef + mu) / 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((EDonor / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))
	tmp = 0
	if EDonor <= -1.52e+159:
		tmp = t_0
	elif EDonor <= -8.2e+73:
		tmp = (NaChar / (1.0 + math.exp((Vef / KbT)))) + (NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / KbT))))
	elif EDonor <= 7.2e+102:
		tmp = (NdChar / (1.0 + math.exp(((Vef + mu) / 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(EDonor / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (EDonor <= -1.52e+159)
		tmp = t_0;
	elseif (EDonor <= -8.2e+73)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)))));
	elseif (EDonor <= 7.2e+102)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Vef + mu) / 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((EDonor / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	tmp = 0.0;
	if (EDonor <= -1.52e+159)
		tmp = t_0;
	elseif (EDonor <= -8.2e+73)
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT))));
	elseif (EDonor <= 7.2e+102)
		tmp = (NdChar / (1.0 + exp(((Vef + mu) / 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[(EDonor / 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[EDonor, -1.52e+159], t$95$0, If[LessEqual[EDonor, -8.2e+73], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 7.2e+102], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(Vef + mu), $MachinePrecision] / 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 EDonor < -1.5199999999999999e159 or 7.2000000000000003e102 < EDonor

   1. Initial program 99.9%

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

   2. 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 69.2

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

   4. Applied rewrites 69.2%

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

   if -1.5199999999999999e159 < EDonor < -8.1999999999999996e73

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 65.3

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

   7. Applied rewrites 65.3%

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

   if -8.1999999999999996e73 < EDonor < 7.2000000000000003e102

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in EDonor around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 43.3

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in Ec around 0

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

      1. lift-+.f64 39.3

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

   10. Applied rewrites 39.3%

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

   11. Taylor expanded in EAccept around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lift-+.f64 76.3

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

   13. Applied rewrites 76.3%

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

Alternative 4: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{Vef + mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}}\\ \mathbf{if}\;mu \leq -1.1 \cdot 10^{-93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;mu \leq 6.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{elif}\;mu \leq 38000000000000:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + t\_0\\ \mathbf{elif}\;mu \leq 7.2 \cdot 10^{+204}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ Vef mu) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ Ev Vef) mu) KbT)))))))
   (if (<= mu -1.1e-93)
     t_1
     (if (<= mu 6.2e-113)
       (+ (/ NaChar (+ 1.0 (exp (/ Vef KbT)))) t_0)
       (if (<= mu 38000000000000.0)
         (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) t_0)
         (if (<= mu 7.2e+204)
           (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) 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((((EDonor + Vef) - Ec) / KbT)));
	double t_1 = (NdChar / (1.0 + exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	double tmp;
	if (mu <= -1.1e-93) {
		tmp = t_1;
	} else if (mu <= 6.2e-113) {
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + t_0;
	} else if (mu <= 38000000000000.0) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + t_0;
	} else if (mu <= 7.2e+204) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = 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) :: tmp
    t_0 = ndchar / (1.0d0 + exp((((edonor + vef) - ec) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp(((vef + mu) / kbt)))) + (nachar / (1.0d0 + exp((((ev + vef) - mu) / kbt))))
    if (mu <= (-1.1d-93)) then
        tmp = t_1
    else if (mu <= 6.2d-113) then
        tmp = (nachar / (1.0d0 + exp((vef / kbt)))) + t_0
    else if (mu <= 38000000000000.0d0) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + t_0
    else if (mu <= 7.2d+204) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + Math.exp((((Ev + Vef) - mu) / KbT))));
	double tmp;
	if (mu <= -1.1e-93) {
		tmp = t_1;
	} else if (mu <= 6.2e-113) {
		tmp = (NaChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	} else if (mu <= 38000000000000.0) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + t_0;
	} else if (mu <= 7.2e+204) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + math.exp((((Ev + Vef) - mu) / KbT))))
	tmp = 0
	if mu <= -1.1e-93:
		tmp = t_1
	elif mu <= 6.2e-113:
		tmp = (NaChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	elif mu <= 38000000000000.0:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + t_0
	elif mu <= 7.2e+204:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Vef + mu) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Ev + Vef) - mu) / KbT)))))
	tmp = 0.0
	if (mu <= -1.1e-93)
		tmp = t_1;
	elseif (mu <= 6.2e-113)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0);
	elseif (mu <= 38000000000000.0)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + t_0);
	elseif (mu <= 7.2e+204)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT)));
	t_1 = (NdChar / (1.0 + exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	tmp = 0.0;
	if (mu <= -1.1e-93)
		tmp = t_1;
	elseif (mu <= 6.2e-113)
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + t_0;
	elseif (mu <= 38000000000000.0)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + t_0;
	elseif (mu <= 7.2e+204)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(Vef + mu), $MachinePrecision] / 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]}, If[LessEqual[mu, -1.1e-93], t$95$1, If[LessEqual[mu, 6.2e-113], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[mu, 38000000000000.0], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[mu, 7.2e+204], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -1.09999999999999998e-93 or 7.2000000000000005e204 < mu

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in EDonor around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 43.3

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in Ec around 0

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

      1. lift-+.f64 39.3

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

   10. Applied rewrites 39.3%

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

   11. Taylor expanded in EAccept around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lift-+.f64 76.3

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

   13. Applied rewrites 76.3%

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

   if -1.09999999999999998e-93 < mu < 6.20000000000000024e-113

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 65.3

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

   7. Applied rewrites 65.3%

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

   if 6.20000000000000024e-113 < mu < 3.8e13

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if 3.8e13 < mu < 7.2000000000000005e204

   1. Initial program 99.9%

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

   2. 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. lift-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

Alternative 5: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ \mathbf{if}\;Ev \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Ev \leq -8.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef + mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))))
   (if (<= Ev -1.75e-6)
     (+ t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
     (if (<= Ev -8.6e-231)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ Vef mu) KbT))))
        (/ NaChar (+ 1.0 (exp (/ (- (+ Ev Vef) mu) KbT)))))
       (+ t_0 (/ NaChar (+ 1.0 (exp (/ 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 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double tmp;
	if (Ev <= -1.75e-6) {
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (Ev <= -8.6e-231) {
		tmp = (NdChar / (1.0 + exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + exp((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) :: tmp
    t_0 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    if (ev <= (-1.75d-6)) then
        tmp = t_0 + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (ev <= (-8.6d-231)) then
        tmp = (ndchar / (1.0d0 + exp(((vef + mu) / kbt)))) + (nachar / (1.0d0 + exp((((ev + vef) - mu) / kbt))))
    else
        tmp = t_0 + (nachar / (1.0d0 + exp((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 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double tmp;
	if (Ev <= -1.75e-6) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (Ev <= -8.6e-231) {
		tmp = (NdChar / (1.0 + Math.exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + Math.exp((((Ev + Vef) - mu) / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((EAccept / 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)))
	tmp = 0
	if Ev <= -1.75e-6:
		tmp = t_0 + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif Ev <= -8.6e-231:
		tmp = (NdChar / (1.0 + math.exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + math.exp((((Ev + Vef) - mu) / KbT))))
	else:
		tmp = t_0 + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	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))))
	tmp = 0.0
	if (Ev <= -1.75e-6)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (Ev <= -8.6e-231)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Vef + mu) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Ev + Vef) - mu) / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / 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)));
	tmp = 0.0;
	if (Ev <= -1.75e-6)
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (Ev <= -8.6e-231)
		tmp = (NdChar / (1.0 + exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	else
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / KbT))));
	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]}, If[LessEqual[Ev, -1.75e-6], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, -8.6e-231], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(Vef + mu), $MachinePrecision] / 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], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Ev < -1.74999999999999997e-6

   1. Initial program 99.9%

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

   2. 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 69.2

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

   4. Applied rewrites 69.2%

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

   if -1.74999999999999997e-6 < Ev < -8.59999999999999996e-231

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in EDonor around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 43.3

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in Ec around 0

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

      1. lift-+.f64 39.3

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

   10. Applied rewrites 39.3%

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

   11. Taylor expanded in EAccept around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lift-+.f64 76.3

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

   13. Applied rewrites 76.3%

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

   if -8.59999999999999996e-231 < Ev

   1. Initial program 99.9%

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

   2. 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: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -3 \cdot 10^{+120}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ \mathbf{elif}\;Ev \leq -8.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef + mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -3e+120)
   (+
    (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
    (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) KbT)))))
   (if (<= Ev -8.6e-231)
     (+
      (/ NdChar (+ 1.0 (exp (/ (+ Vef mu) KbT))))
      (/ NaChar (+ 1.0 (exp (/ (- (+ Ev Vef) mu) KbT)))))
     (+
      (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
      (/ NaChar (+ 1.0 (exp (/ EAccept 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 (Ev <= -3e+120) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT))));
	} else if (Ev <= -8.6e-231) {
		tmp = (NdChar / (1.0 + exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	} else {
		tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((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) :: tmp
    if (ev <= (-3d+120)) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp((((edonor + vef) - ec) / kbt))))
    else if (ev <= (-8.6d-231)) then
        tmp = (ndchar / (1.0d0 + exp(((vef + mu) / kbt)))) + (nachar / (1.0d0 + exp((((ev + vef) - mu) / kbt))))
    else
        tmp = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp((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 tmp;
	if (Ev <= -3e+120) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / KbT))));
	} else if (Ev <= -8.6e-231) {
		tmp = (NdChar / (1.0 + Math.exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + Math.exp((((Ev + Vef) - mu) / KbT))));
	} else {
		tmp = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -3e+120:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / KbT))))
	elif Ev <= -8.6e-231:
		tmp = (NdChar / (1.0 + math.exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + math.exp((((Ev + Vef) - mu) / KbT))))
	else:
		tmp = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -3e+120)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)))));
	elseif (Ev <= -8.6e-231)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Vef + mu) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Ev + Vef) - mu) / KbT)))));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -3e+120)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT))));
	elseif (Ev <= -8.6e-231)
		tmp = (NdChar / (1.0 + exp(((Vef + mu) / KbT)))) + (NaChar / (1.0 + exp((((Ev + Vef) - mu) / KbT))));
	else
		tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -3e+120], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, -8.6e-231], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(Vef + mu), $MachinePrecision] / 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], 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[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Ev < -3e120

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if -3e120 < Ev < -8.59999999999999996e-231

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in EDonor around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 43.3

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in Ec around 0

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

      1. lift-+.f64 39.3

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

   10. Applied rewrites 39.3%

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

   11. Taylor expanded in EAccept around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lift-+.f64 76.3

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

   13. Applied rewrites 76.3%

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

   if -8.59999999999999996e-231 < Ev

   1. Initial program 99.9%

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

   2. 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 7: 72.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ \mathbf{if}\;Ev \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + t\_0\\ \mathbf{elif}\;Ev \leq -4.9 \cdot 10^{-231}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) KbT))))))
   (if (<= Ev -2e+120)
     (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) t_0)
     (if (<= Ev -4.9e-231)
       (+ (/ NaChar (+ 1.0 (exp (/ Vef KbT)))) t_0)
       (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT)));
	double tmp;
	if (Ev <= -2e+120) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + t_0;
	} else if (Ev <= -4.9e-231) {
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + t_0;
	} else {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + 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((((edonor + vef) - ec) / kbt)))
    if (ev <= (-2d+120)) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + t_0
    else if (ev <= (-4.9d-231)) then
        tmp = (nachar / (1.0d0 + exp((vef / kbt)))) + t_0
    else
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / KbT)));
	double tmp;
	if (Ev <= -2e+120) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + t_0;
	} else if (Ev <= -4.9e-231) {
		tmp = (NaChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	} else {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / KbT)))
	tmp = 0
	if Ev <= -2e+120:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + t_0
	elif Ev <= -4.9e-231:
		tmp = (NaChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	else:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT))))
	tmp = 0.0
	if (Ev <= -2e+120)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + t_0);
	elseif (Ev <= -4.9e-231)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0);
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT)));
	tmp = 0.0;
	if (Ev <= -2e+120)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + t_0;
	elseif (Ev <= -4.9e-231)
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + t_0;
	else
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ev, -2e+120], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[Ev, -4.9e-231], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Ev < -2e120

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if -2e120 < Ev < -4.90000000000000003e-231

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 65.3

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

   7. Applied rewrites 65.3%

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

   if -4.90000000000000003e-231 < Ev

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 64.2

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

   10. Applied rewrites 64.2%

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

Alternative 8: 70.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4.5 \cdot 10^{-274}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{elif}\;t\_1 \leq 10^{-11}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))
          (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) KbT))))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -2e-300)
     t_0
     (if (<= t_1 4.5e-274)
       (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))
       (if (<= t_1 1e-11)
         (+
          (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
          (/ NdChar (+ 1.0 (exp (/ (- EDonor Ec) KbT)))))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / 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 <= -2e-300) {
		tmp = t_0;
	} else if (t_1 <= 4.5e-274) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else if (t_1 <= 1e-11) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp(((EDonor - Ec) / 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) :: t_1
    real(8) :: tmp
    t_0 = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar / (1.0d0 + exp((((edonor + vef) - ec) / kbt))))
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (t_1 <= (-2d-300)) then
        tmp = t_0
    else if (t_1 <= 4.5d-274) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else if (t_1 <= 1d-11) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp(((edonor - ec) / kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / 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 <= -2e-300) {
		tmp = t_0;
	} else if (t_1 <= 4.5e-274) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else if (t_1 <= 1e-11) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp(((EDonor - Ec) / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / 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 <= -2e-300:
		tmp = t_0
	elif t_1 <= 4.5e-274:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	elif t_1 <= 1e-11:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp(((EDonor - Ec) / KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / 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 <= -2e-300)
		tmp = t_0;
	elseif (t_1 <= 4.5e-274)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	elseif (t_1 <= 1e-11)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Ec) / KbT)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / 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 <= -2e-300)
		tmp = t_0;
	elseif (t_1 <= 4.5e-274)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	elseif (t_1 <= 1e-11)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp(((EDonor - Ec) / KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $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, -2e-300], t$95$0, If[LessEqual[t$95$1, 4.5e-274], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-11], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.00000000000000005e-300 or 9.99999999999999939e-12 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 64.2

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

   10. Applied rewrites 64.2%

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

   if -2.00000000000000005e-300 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.49999999999999991e-274

   1. Initial program 99.9%

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

   2. 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. lift-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

   if 4.49999999999999991e-274 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.99999999999999939e-12

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in Vef around 0

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

   10. Applied rewrites 58.0%

       \[\leadsto \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ \mathbf{if}\;Ev \leq -0.00091:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + t\_0\\ \mathbf{elif}\;Ev \leq -2.1 \cdot 10^{-191}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) KbT))))))
   (if (<= Ev -0.00091)
     (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) t_0)
     (if (<= Ev -2.1e-191)
       (+
        (/ NaChar (+ 1.0 (exp (/ Vef KbT))))
        (/ NdChar (+ 1.0 (exp (/ (- EDonor Ec) KbT)))))
       (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT)));
	double tmp;
	if (Ev <= -0.00091) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + t_0;
	} else if (Ev <= -2.1e-191) {
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar / (1.0 + exp(((EDonor - Ec) / KbT))));
	} else {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + 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((((edonor + vef) - ec) / kbt)))
    if (ev <= (-0.00091d0)) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + t_0
    else if (ev <= (-2.1d-191)) then
        tmp = (nachar / (1.0d0 + exp((vef / kbt)))) + (ndchar / (1.0d0 + exp(((edonor - ec) / kbt))))
    else
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / KbT)));
	double tmp;
	if (Ev <= -0.00091) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + t_0;
	} else if (Ev <= -2.1e-191) {
		tmp = (NaChar / (1.0 + Math.exp((Vef / KbT)))) + (NdChar / (1.0 + Math.exp(((EDonor - Ec) / KbT))));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / KbT)))
	tmp = 0
	if Ev <= -0.00091:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + t_0
	elif Ev <= -2.1e-191:
		tmp = (NaChar / (1.0 + math.exp((Vef / KbT)))) + (NdChar / (1.0 + math.exp(((EDonor - Ec) / KbT))))
	else:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT))))
	tmp = 0.0
	if (Ev <= -0.00091)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + t_0);
	elseif (Ev <= -2.1e-191)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Ec) / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT)));
	tmp = 0.0;
	if (Ev <= -0.00091)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + t_0;
	elseif (Ev <= -2.1e-191)
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar / (1.0 + exp(((EDonor - Ec) / KbT))));
	else
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ev, -0.00091], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[Ev, -2.1e-191], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Ev < -9.1e-4

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if -9.1e-4 < Ev < -2.09999999999999985e-191

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Vef around inf

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

      1. lower-/.f64 65.3

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in Vef around 0

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

   10. Applied rewrites 56.4%

       \[\leadsto \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}} \]

   if -2.09999999999999985e-191 < Ev

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 64.2

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

   10. Applied rewrites 64.2%

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

Alternative 10: 66.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4.5 \cdot 10^{-274}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
          (/ NdChar (+ 1.0 (exp (/ (- EDonor Ec) KbT))))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -2e-277)
     t_0
     (if (<= t_1 4.5e-274)
       (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp(((EDonor - Ec) / 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 <= -2e-277) {
		tmp = t_0;
	} else if (t_1 <= 4.5e-274) {
		tmp = NaChar / (1.0 + exp((((EAccept + (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) :: t_1
    real(8) :: tmp
    t_0 = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp(((edonor - ec) / kbt))))
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (t_1 <= (-2d-277)) then
        tmp = t_0
    else if (t_1 <= 4.5d-274) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp(((EDonor - Ec) / 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 <= -2e-277) {
		tmp = t_0;
	} else if (t_1 <= 4.5e-274) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp(((EDonor - Ec) / 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 <= -2e-277:
		tmp = t_0
	elif t_1 <= 4.5e-274:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Ec) / 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 <= -2e-277)
		tmp = t_0;
	elseif (t_1 <= 4.5e-274)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp(((EDonor - Ec) / 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 <= -2e-277)
		tmp = t_0;
	elseif (t_1 <= 4.5e-274)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - Ec), $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, -2e-277], t$95$0, If[LessEqual[t$95$1, 4.5e-274], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))))) < -1.99999999999999994e-277 or 4.49999999999999991e-274 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in Vef around 0

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

   10. Applied rewrites 58.0%

       \[\leadsto \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}} \]

   if -1.99999999999999994e-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))))) < 4.49999999999999991e-274

   1. Initial program 99.9%

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

   2. 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. lift-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

Alternative 11: 66.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ t_2 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ t_3 := t\_0 + t\_2\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot NdChar + t\_2\\ \mathbf{elif}\;t\_3 \leq 4.5 \cdot 10^{-274}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{-66}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 + 0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))
        (t_2
         (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))
        (t_3 (+ t_0 t_2)))
   (if (<= t_3 -1e-132)
     (+ (* 0.5 NdChar) t_2)
     (if (<= t_3 4.5e-274)
       t_1
       (if (<= t_3 1e-66)
         (+
          (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
         (if (<= t_3 2e+111) t_1 (+ t_0 (* 0.5 NaChar))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	double t_2 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT)));
	double t_3 = t_0 + t_2;
	double tmp;
	if (t_3 <= -1e-132) {
		tmp = (0.5 * NdChar) + t_2;
	} else if (t_3 <= 4.5e-274) {
		tmp = t_1;
	} else if (t_3 <= 1e-66) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (t_3 <= 2e+111) {
		tmp = t_1;
	} else {
		tmp = t_0 + (0.5 * NaChar);
	}
	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 = ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))
    t_1 = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    t_2 = nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt)))
    t_3 = t_0 + t_2
    if (t_3 <= (-1d-132)) then
        tmp = (0.5d0 * ndchar) + t_2
    else if (t_3 <= 4.5d-274) then
        tmp = t_1
    else if (t_3 <= 1d-66) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (t_3 <= 2d+111) then
        tmp = t_1
    else
        tmp = t_0 + (0.5d0 * nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	double t_2 = NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT)));
	double t_3 = t_0 + t_2;
	double tmp;
	if (t_3 <= -1e-132) {
		tmp = (0.5 * NdChar) + t_2;
	} else if (t_3 <= 4.5e-274) {
		tmp = t_1;
	} else if (t_3 <= 1e-66) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (t_3 <= 2e+111) {
		tmp = t_1;
	} else {
		tmp = t_0 + (0.5 * NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	t_2 = NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT)))
	t_3 = t_0 + t_2
	tmp = 0
	if t_3 <= -1e-132:
		tmp = (0.5 * NdChar) + t_2
	elif t_3 <= 4.5e-274:
		tmp = t_1
	elif t_3 <= 1e-66:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif t_3 <= 2e+111:
		tmp = t_1
	else:
		tmp = t_0 + (0.5 * NaChar)
	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(EAccept + Float64(Ev + Vef)) - mu) / KbT))))
	t_2 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT))))
	t_3 = Float64(t_0 + t_2)
	tmp = 0.0
	if (t_3 <= -1e-132)
		tmp = Float64(Float64(0.5 * NdChar) + t_2);
	elseif (t_3 <= 4.5e-274)
		tmp = t_1;
	elseif (t_3 <= 1e-66)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (t_3 <= 2e+111)
		tmp = t_1;
	else
		tmp = Float64(t_0 + Float64(0.5 * NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	t_2 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT)));
	t_3 = t_0 + t_2;
	tmp = 0.0;
	if (t_3 <= -1e-132)
		tmp = (0.5 * NdChar) + t_2;
	elseif (t_3 <= 4.5e-274)
		tmp = t_1;
	elseif (t_3 <= 1e-66)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (t_3 <= 2e+111)
		tmp = t_1;
	else
		tmp = t_0 + (0.5 * NaChar);
	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[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(t$95$0 + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-132], N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 4.5e-274], t$95$1, If[LessEqual[t$95$3, 1e-66], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+111], t$95$1, N[(t$95$0 + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 99.9%

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

   2. 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 47.3

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

   4. Applied rewrites 47.3%

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

   if -9.9999999999999999e-133 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.49999999999999991e-274 or 9.9999999999999998e-67 < (+.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))))) < 1.99999999999999991e111

   1. Initial program 99.9%

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

   2. 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. lift-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

   if 4.49999999999999991e-274 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999998e-67

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in Ev around inf

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in EDonor around inf

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

      1. lower-/.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if 1.99999999999999991e111 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

Alternative 12: 65.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ t_2 := t\_0 + t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot NdChar + t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + 0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1
         (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))
        (t_2 (+ t_0 t_1)))
   (if (<= t_2 -1e-132)
     (+ (* 0.5 NdChar) t_1)
     (if (<= t_2 2e+111)
       (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))
       (+ t_0 (* 0.5 NaChar))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double 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 <= -1e-132) {
		tmp = (0.5 * NdChar) + t_1;
	} else if (t_2 <= 2e+111) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_0 + (0.5 * NaChar);
	}
	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 <= (-1d-132)) then
        tmp = (0.5d0 * ndchar) + t_1
    else if (t_2 <= 2d+111) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = t_0 + (0.5d0 * nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double 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 <= -1e-132) {
		tmp = (0.5 * NdChar) + t_1;
	} else if (t_2 <= 2e+111) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_0 + (0.5 * NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT)))
	t_2 = t_0 + t_1
	tmp = 0
	if t_2 <= -1e-132:
		tmp = (0.5 * NdChar) + t_1
	elif t_2 <= 2e+111:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = t_0 + (0.5 * NaChar)
	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 <= -1e-132)
		tmp = Float64(Float64(0.5 * NdChar) + t_1);
	elseif (t_2 <= 2e+111)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = Float64(t_0 + Float64(0.5 * NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT)));
	t_2 = t_0 + t_1;
	tmp = 0.0;
	if (t_2 <= -1e-132)
		tmp = (0.5 * NdChar) + t_1;
	elseif (t_2 <= 2e+111)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = t_0 + (0.5 * NaChar);
	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, -1e-132], N[(N[(0.5 * NdChar), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+111], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(0.5 * NaChar), $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))))) < -9.9999999999999999e-133

   1. Initial program 99.9%

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

   2. 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 47.3

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

   4. Applied rewrites 47.3%

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

   if -9.9999999999999999e-133 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1.99999999999999991e111

   1. Initial program 99.9%

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

   2. 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. lift-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

   if 1.99999999999999991e111 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

Alternative 13: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := t\_0 + 0.5 \cdot NaChar\\ t_2 := t\_0 + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1 (+ t_0 (* 0.5 NaChar)))
        (t_2
         (+
          t_0
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_2 -1e+152)
     t_1
     (if (<= t_2 2e+111)
       (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) 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 = t_0 + (0.5 * NaChar);
	double t_2 = t_0 + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_2 <= -1e+152) {
		tmp = t_1;
	} else if (t_2 <= 2e+111) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = 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 = t_0 + (0.5d0 * nachar)
    t_2 = t_0 + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (t_2 <= (-1d+152)) then
        tmp = t_1
    else if (t_2 <= 2d+111) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = t_0 + (0.5 * NaChar);
	double t_2 = t_0 + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_2 <= -1e+152) {
		tmp = t_1;
	} else if (t_2 <= 2e+111) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))
	t_1 = t_0 + (0.5 * NaChar)
	t_2 = t_0 + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))
	tmp = 0
	if t_2 <= -1e+152:
		tmp = t_1
	elif t_2 <= 2e+111:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_1 = Float64(t_0 + Float64(0.5 * NaChar))
	t_2 = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_2 <= -1e+152)
		tmp = t_1;
	elseif (t_2 <= 2e+111)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	t_1 = t_0 + (0.5 * NaChar);
	t_2 = t_0 + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	tmp = 0.0;
	if (t_2 <= -1e+152)
		tmp = t_1;
	elseif (t_2 <= 2e+111)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+152], t$95$1, If[LessEqual[t$95$2, 2e+111], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1e152 or 1.99999999999999991e111 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   if -1e152 < (+.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))))) < 1.99999999999999991e111

   1. Initial program 99.9%

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

   2. 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. lift-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

Alternative 14: 63.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{\left(Vef + mu\right) - Ec}{KbT}}} + 0.5 \cdot NaChar\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+ (/ NdChar (+ 1.0 (exp (/ (- (+ Vef mu) Ec) KbT)))) (* 0.5 NaChar)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -1e+152)
     t_0
     (if (<= t_1 2e+113)
       (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((((Vef + mu) - Ec) / KbT)))) + (0.5 * NaChar);
	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 <= -1e+152) {
		tmp = t_0;
	} else if (t_1 <= 2e+113) {
		tmp = NaChar / (1.0 + exp((((EAccept + (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) :: t_1
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((((vef + mu) - ec) / kbt)))) + (0.5d0 * nachar)
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (t_1 <= (-1d+152)) then
        tmp = t_0
    else if (t_1 <= 2d+113) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((((Vef + mu) - Ec) / KbT)))) + (0.5 * NaChar);
	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 <= -1e+152) {
		tmp = t_0;
	} else if (t_1 <= 2e+113) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((((Vef + mu) - Ec) / KbT)))) + (0.5 * NaChar)
	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 <= -1e+152:
		tmp = t_0
	elif t_1 <= 2e+113:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + mu) - Ec) / KbT)))) + Float64(0.5 * NaChar))
	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 <= -1e+152)
		tmp = t_0;
	elseif (t_1 <= 2e+113)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((((Vef + mu) - Ec) / KbT)))) + (0.5 * NaChar);
	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 <= -1e+152)
		tmp = t_0;
	elseif (t_1 <= 2e+113)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $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, -1e+152], t$95$0, If[LessEqual[t$95$1, 2e+113], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))))) < -1e152 or 2e113 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in EDonor around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 43.3

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

   7. Applied rewrites 43.3%

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

   if -1e152 < (+.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))))) < 2e113

   1. Initial program 99.9%

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

   2. 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. lift-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

Alternative 15: 63.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu - Ec}{KbT}}} + 0.5 \cdot NaChar\\ \mathbf{elif}\;KbT \leq 2.8 \cdot 10^{+204}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\frac{\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec}{KbT}} + 0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4.5e+164)
   (+ (/ NdChar (+ 1.0 (exp (/ (- mu Ec) KbT)))) (* 0.5 NaChar))
   (if (<= KbT 2.8e+204)
     (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))
     (+
      (/ NdChar (/ (- (+ EDonor (+ Vef (+ mu (* 2.0 KbT)))) Ec) KbT))
      (* 0.5 NaChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.5e+164) {
		tmp = (NdChar / (1.0 + exp(((mu - Ec) / KbT)))) + (0.5 * NaChar);
	} else if (KbT <= 2.8e+204) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar);
	}
	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 <= (-4.5d+164)) then
        tmp = (ndchar / (1.0d0 + exp(((mu - ec) / kbt)))) + (0.5d0 * nachar)
    else if (kbt <= 2.8d+204) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = (ndchar / (((edonor + (vef + (mu + (2.0d0 * kbt)))) - ec) / kbt)) + (0.5d0 * nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.5e+164) {
		tmp = (NdChar / (1.0 + Math.exp(((mu - Ec) / KbT)))) + (0.5 * NaChar);
	} else if (KbT <= 2.8e+204) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -4.5e+164:
		tmp = (NdChar / (1.0 + math.exp(((mu - Ec) / KbT)))) + (0.5 * NaChar)
	elif KbT <= 2.8e+204:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4.5e+164)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Ec) / KbT)))) + Float64(0.5 * NaChar));
	elseif (KbT <= 2.8e+204)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = Float64(Float64(NdChar / Float64(Float64(Float64(EDonor + Float64(Vef + Float64(mu + Float64(2.0 * KbT)))) - Ec) / KbT)) + Float64(0.5 * NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -4.5e+164)
		tmp = (NdChar / (1.0 + exp(((mu - Ec) / KbT)))) + (0.5 * NaChar);
	elseif (KbT <= 2.8e+204)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4.5e+164], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.8e+204], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[(N[(EDonor + N[(Vef + N[(mu + N[(2.0 * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.49999999999999975e164

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in EDonor around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 43.3

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in Vef around 0

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

   10. Applied rewrites 39.7%

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

   if -4.49999999999999975e164 < KbT < 2.80000000000000025e204

   1. Initial program 99.9%

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

   2. 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. lift-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

   if 2.80000000000000025e204 < KbT

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. div-add-rev N/A

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

      4. div-add N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 25.8

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in KbT around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 26.1

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

   10. Applied rewrites 26.1%

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

Alternative 16: 62.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + 0.5 \cdot NaChar\\ \mathbf{elif}\;KbT \leq 2.8 \cdot 10^{+204}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\frac{\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec}{KbT}} + 0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4.5e+164)
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* 0.5 NaChar))
   (if (<= KbT 2.8e+204)
     (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))
     (+
      (/ NdChar (/ (- (+ EDonor (+ Vef (+ mu (* 2.0 KbT)))) Ec) KbT))
      (* 0.5 NaChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.5e+164) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (0.5 * NaChar);
	} else if (KbT <= 2.8e+204) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar);
	}
	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 <= (-4.5d+164)) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (0.5d0 * nachar)
    else if (kbt <= 2.8d+204) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = (ndchar / (((edonor + (vef + (mu + (2.0d0 * kbt)))) - ec) / kbt)) + (0.5d0 * nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.5e+164) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (0.5 * NaChar);
	} else if (KbT <= 2.8e+204) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -4.5e+164:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (0.5 * NaChar)
	elif KbT <= 2.8e+204:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4.5e+164)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(0.5 * NaChar));
	elseif (KbT <= 2.8e+204)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = Float64(Float64(NdChar / Float64(Float64(Float64(EDonor + Float64(Vef + Float64(mu + Float64(2.0 * KbT)))) - Ec) / KbT)) + Float64(0.5 * NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -4.5e+164)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (0.5 * NaChar);
	elseif (KbT <= 2.8e+204)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4.5e+164], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.8e+204], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[(N[(EDonor + N[(Vef + N[(mu + N[(2.0 * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.49999999999999975e164

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in mu around inf

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

      1. lower-/.f64 35.2

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

   7. Applied rewrites 35.2%

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

   if -4.49999999999999975e164 < KbT < 2.80000000000000025e204

   1. Initial program 99.9%

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

   2. 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. lift-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

   if 2.80000000000000025e204 < KbT

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. div-add-rev N/A

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

      4. div-add N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 25.8

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in KbT around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 26.1

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

   10. Applied rewrites 26.1%

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

Alternative 17: 57.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + 0.5 \cdot NaChar\\ \mathbf{elif}\;KbT \leq 4.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + Vef\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\frac{\left(EDonor + \left(Vef + \left(mu + 2 \cdot KbT\right)\right)\right) - Ec}{KbT}} + 0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4.5e+164)
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* 0.5 NaChar))
   (if (<= KbT 4.5e+203)
     (/ NaChar (+ 1.0 (exp (/ (+ EAccept (+ Ev Vef)) KbT))))
     (+
      (/ NdChar (/ (- (+ EDonor (+ Vef (+ mu (* 2.0 KbT)))) Ec) KbT))
      (* 0.5 NaChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.5e+164) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (0.5 * NaChar);
	} else if (KbT <= 4.5e+203) {
		tmp = NaChar / (1.0 + exp(((EAccept + (Ev + Vef)) / KbT)));
	} else {
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar);
	}
	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 <= (-4.5d+164)) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (0.5d0 * nachar)
    else if (kbt <= 4.5d+203) then
        tmp = nachar / (1.0d0 + exp(((eaccept + (ev + vef)) / kbt)))
    else
        tmp = (ndchar / (((edonor + (vef + (mu + (2.0d0 * kbt)))) - ec) / kbt)) + (0.5d0 * nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.5e+164) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (0.5 * NaChar);
	} else if (KbT <= 4.5e+203) {
		tmp = NaChar / (1.0 + Math.exp(((EAccept + (Ev + Vef)) / KbT)));
	} else {
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -4.5e+164:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (0.5 * NaChar)
	elif KbT <= 4.5e+203:
		tmp = NaChar / (1.0 + math.exp(((EAccept + (Ev + Vef)) / KbT)))
	else:
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4.5e+164)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(0.5 * NaChar));
	elseif (KbT <= 4.5e+203)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Float64(Ev + Vef)) / KbT))));
	else
		tmp = Float64(Float64(NdChar / Float64(Float64(Float64(EDonor + Float64(Vef + Float64(mu + Float64(2.0 * KbT)))) - Ec) / KbT)) + Float64(0.5 * NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -4.5e+164)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (0.5 * NaChar);
	elseif (KbT <= 4.5e+203)
		tmp = NaChar / (1.0 + exp(((EAccept + (Ev + Vef)) / KbT)));
	else
		tmp = (NdChar / (((EDonor + (Vef + (mu + (2.0 * KbT)))) - Ec) / KbT)) + (0.5 * NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4.5e+164], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.5e+203], N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[(N[(EDonor + N[(Vef + N[(mu + N[(2.0 * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.49999999999999975e164

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in mu around inf

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

      1. lower-/.f64 35.2

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

   7. Applied rewrites 35.2%

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

   if -4.49999999999999975e164 < KbT < 4.5000000000000003e203

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 53.9

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

   7. Applied rewrites 53.9%

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

   if 4.5000000000000003e203 < KbT

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. div-add-rev N/A

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

      4. div-add N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 25.8

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in KbT around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 26.1

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

   10. Applied rewrites 26.1%

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

Alternative 18: 57.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + 0.5 \cdot NaChar\\ \mathbf{elif}\;KbT \leq 4.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + Vef\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4.5e+164)
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* 0.5 NaChar))
   (if (<= KbT 4.5e+203)
     (/ NaChar (+ 1.0 (exp (/ (+ EAccept (+ Ev Vef)) KbT))))
     (fma 0.5 NaChar (* 0.5 NdChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.5e+164) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (0.5 * NaChar);
	} else if (KbT <= 4.5e+203) {
		tmp = NaChar / (1.0 + exp(((EAccept + (Ev + Vef)) / KbT)));
	} else {
		tmp = fma(0.5, NaChar, (0.5 * NdChar));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4.5e+164)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(0.5 * NaChar));
	elseif (KbT <= 4.5e+203)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Float64(Ev + Vef)) / KbT))));
	else
		tmp = fma(0.5, NaChar, Float64(0.5 * NdChar));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4.5e+164], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.5e+203], N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * NaChar + N[(0.5 * NdChar), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.49999999999999975e164

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in mu around inf

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

      1. lower-/.f64 35.2

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

   7. Applied rewrites 35.2%

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

   if -4.49999999999999975e164 < KbT < 4.5000000000000003e203

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 53.9

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

   7. Applied rewrites 53.9%

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

   if 4.5000000000000003e203 < KbT

   1. Initial program 99.9%

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

   2. 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 27.7

         \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

   4. Applied rewrites 27.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 57.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4.6 \cdot 10^{+164}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{EDonor + \left(Vef + mu\right)}{KbT}} + 0.5 \cdot NaChar\\ \mathbf{elif}\;KbT \leq 4.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + Vef\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4.6e+164)
   (+ (/ NdChar (+ 2.0 (/ (+ EDonor (+ Vef mu)) KbT))) (* 0.5 NaChar))
   (if (<= KbT 4.5e+203)
     (/ NaChar (+ 1.0 (exp (/ (+ EAccept (+ Ev Vef)) KbT))))
     (fma 0.5 NaChar (* 0.5 NdChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.6e+164) {
		tmp = (NdChar / (2.0 + ((EDonor + (Vef + mu)) / KbT))) + (0.5 * NaChar);
	} else if (KbT <= 4.5e+203) {
		tmp = NaChar / (1.0 + exp(((EAccept + (Ev + Vef)) / KbT)));
	} else {
		tmp = fma(0.5, NaChar, (0.5 * NdChar));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4.6e+164)
		tmp = Float64(Float64(NdChar / Float64(2.0 + Float64(Float64(EDonor + Float64(Vef + mu)) / KbT))) + Float64(0.5 * NaChar));
	elseif (KbT <= 4.5e+203)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Float64(Ev + Vef)) / KbT))));
	else
		tmp = fma(0.5, NaChar, Float64(0.5 * NdChar));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4.6e+164], N[(N[(NdChar / N[(2.0 + N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.5e+203], N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * NaChar + N[(0.5 * NdChar), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.5999999999999999e164

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. div-add-rev N/A

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

      4. div-add N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 25.8

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in Ec around 0

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

      1. div-add-rev N/A

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

      2. div-add N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 26.0

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

   10. Applied rewrites 26.0%

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

   if -4.5999999999999999e164 < KbT < 4.5000000000000003e203

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 53.9

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

   7. Applied rewrites 53.9%

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

   if 4.5000000000000003e203 < KbT

   1. Initial program 99.9%

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

   2. 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 27.7

         \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

   4. Applied rewrites 27.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 50.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4.6 \cdot 10^{+164}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{EDonor + \left(Vef + mu\right)}{KbT}} + 0.5 \cdot NaChar\\ \mathbf{elif}\;KbT \leq 2 \cdot 10^{+203}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4.6e+164)
   (+ (/ NdChar (+ 2.0 (/ (+ EDonor (+ Vef mu)) KbT))) (* 0.5 NaChar))
   (if (<= KbT 2e+203)
     (/ NaChar (+ 1.0 (exp (/ (+ EAccept Vef) KbT))))
     (fma 0.5 NaChar (* 0.5 NdChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -4.6e+164) {
		tmp = (NdChar / (2.0 + ((EDonor + (Vef + mu)) / KbT))) + (0.5 * NaChar);
	} else if (KbT <= 2e+203) {
		tmp = NaChar / (1.0 + exp(((EAccept + Vef) / KbT)));
	} else {
		tmp = fma(0.5, NaChar, (0.5 * NdChar));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4.6e+164)
		tmp = Float64(Float64(NdChar / Float64(2.0 + Float64(Float64(EDonor + Float64(Vef + mu)) / KbT))) + Float64(0.5 * NaChar));
	elseif (KbT <= 2e+203)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Vef) / KbT))));
	else
		tmp = fma(0.5, NaChar, Float64(0.5 * NdChar));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4.6e+164], N[(N[(NdChar / N[(2.0 + N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2e+203], N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + Vef), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * NaChar + N[(0.5 * NdChar), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.5999999999999999e164

   1. Initial program 99.9%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in KbT around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. div-add-rev N/A

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

      4. div-add N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 25.8

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in Ec around 0

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

      1. div-add-rev N/A

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

      2. div-add N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 26.0

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

   10. Applied rewrites 26.0%

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

   if -4.5999999999999999e164 < KbT < 2e203

   1. Initial program 99.9%

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

   2. Taylor expanded in mu around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 53.9

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in Ev around 0

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

      1. lower-+.f64 47.0

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

   10. Applied rewrites 47.0%

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

   if 2e203 < KbT

   1. Initial program 99.9%

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

   2. 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 27.7

         \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

   4. Applied rewrites 27.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 27.7% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. 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 27.7

      \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

4. Applied rewrites 27.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
5. Add Preprocessing

Alternative 22: 22.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -5.2 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{elif}\;NdChar \leq 3.8 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NdChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -5.2e+150)
   (* 0.5 NdChar)
   (if (<= NdChar 3.8e+59) (* 0.5 NaChar) (* 0.5 NdChar))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -5.2e+150) {
		tmp = 0.5 * NdChar;
	} else if (NdChar <= 3.8e+59) {
		tmp = 0.5 * NaChar;
	} else {
		tmp = 0.5 * NdChar;
	}
	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 <= (-5.2d+150)) then
        tmp = 0.5d0 * ndchar
    else if (ndchar <= 3.8d+59) then
        tmp = 0.5d0 * nachar
    else
        tmp = 0.5d0 * ndchar
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -5.2e+150) {
		tmp = 0.5 * NdChar;
	} else if (NdChar <= 3.8e+59) {
		tmp = 0.5 * NaChar;
	} else {
		tmp = 0.5 * NdChar;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -5.2e+150:
		tmp = 0.5 * NdChar
	elif NdChar <= 3.8e+59:
		tmp = 0.5 * NaChar
	else:
		tmp = 0.5 * NdChar
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -5.2e+150)
		tmp = Float64(0.5 * NdChar);
	elseif (NdChar <= 3.8e+59)
		tmp = Float64(0.5 * NaChar);
	else
		tmp = Float64(0.5 * NdChar);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -5.2e+150)
		tmp = 0.5 * NdChar;
	elseif (NdChar <= 3.8e+59)
		tmp = 0.5 * NaChar;
	else
		tmp = 0.5 * NdChar;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -5.2e+150], N[(0.5 * NdChar), $MachinePrecision], If[LessEqual[NdChar, 3.8e+59], N[(0.5 * NaChar), $MachinePrecision], N[(0.5 * NdChar), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -5.20000000000000012e150 or 3.8000000000000001e59 < NdChar

   1. Initial program 99.9%

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

   2. 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 27.7

         \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

   4. Applied rewrites 27.7%

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

   5. Taylor expanded in NdChar around 0

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

      1. lift-*.f64 18.3

         \[\leadsto 0.5 \cdot NaChar \]

   7. Applied rewrites 18.3%

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

   8. Taylor expanded in NdChar around inf

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

      1. lower-*.f64 18.4

         \[\leadsto 0.5 \cdot NdChar \]

   10. Applied rewrites 18.4%

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

   if -5.20000000000000012e150 < NdChar < 3.8000000000000001e59

   1. Initial program 99.9%

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

   2. 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 27.7

         \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

   4. Applied rewrites 27.7%

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

   5. Taylor expanded in NdChar around 0

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

      1. lift-*.f64 18.3

         \[\leadsto 0.5 \cdot NaChar \]

   7. Applied rewrites 18.3%

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

Alternative 23: 18.3% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot NaChar \end{array} \]

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 99.9%

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

2. 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 27.7

      \[\leadsto \mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right) \]

4. Applied rewrites 27.7%

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

5. Taylor expanded in NdChar around 0

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

   1. lift-*.f64 18.3

      \[\leadsto 0.5 \cdot NaChar \]

7. Applied rewrites 18.3%

   \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(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))))))