Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 9.4s

Alternatives: 19

Speedup: N/A×

• could not determine a ground truth

  1. NdChar = 5.760056712694657e-249
  2. Ec = 7.507345143043463e+158
  3. Vef = 3.7047550722273864e+268
  4. EDonor = 9.470464091383871e-281
  5. mu = 2.1670198893663697e+118
  6. KbT = -1.204061875418621e-138
  7. NaChar = -9.16613395752087e-250
  8. Ev = -5.621004051433265e-73
  9. EAccept = -3.5818485856972586e-127

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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-flip N/A

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

   4. associate-/r/ N/A

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

   5. lower-fma.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 2: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;mu \leq -1.6 \cdot 10^{+173}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq 8 \cdot 10^{+128}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + Vef\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= mu -1.6e+173)
     t_0
     (if (<= mu 8e+128)
       (+
        (/ 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((mu / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (mu <= -1.6e+173) {
		tmp = t_0;
	} else if (mu <= 8e+128) {
		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((mu / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (mu <= (-1.6d+173)) then
        tmp = t_0
    else if (mu <= 8d+128) 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((mu / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (mu <= -1.6e+173) {
		tmp = t_0;
	} else if (mu <= 8e+128) {
		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((mu / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))
	tmp = 0
	if mu <= -1.6e+173:
		tmp = t_0
	elif mu <= 8e+128:
		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(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (mu <= -1.6e+173)
		tmp = t_0;
	elseif (mu <= 8e+128)
		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((mu / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	tmp = 0.0;
	if (mu <= -1.6e+173)
		tmp = t_0;
	elseif (mu <= 8e+128)
		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[(mu / 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[mu, -1.6e+173], t$95$0, If[LessEqual[mu, 8e+128], 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 < -1.6000000000000001e173 or 8.0000000000000006e128 < mu

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around inf

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

      1. lower-/.f64 69.5

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

   4. Applied rewrites 69.5%

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

   if -1.6000000000000001e173 < mu < 8.0000000000000006e128

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around 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

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -9.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Ev \leq -3.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + Vef}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -9.2e+120)
   (+
    (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
    (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
   (if (<= Ev -3.7e+40)
     (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))
     (+
      (/ NaChar (+ 1.0 (exp (/ (+ EAccept Vef) KbT))))
      (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) 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 <= -9.2e+120) {
		tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (Ev <= -3.7e+40) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + exp(((EAccept + Vef) / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / 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 <= (-9.2d+120)) then
        tmp = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (ev <= (-3.7d+40)) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = (nachar / (1.0d0 + exp(((eaccept + vef) / kbt)))) + (ndchar / (1.0d0 + exp((((edonor + vef) - ec) / 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 <= -9.2e+120) {
		tmp = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (Ev <= -3.7e+40) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((EAccept + Vef) / KbT)))) + (NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -9.2e+120:
		tmp = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif Ev <= -3.7e+40:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = (NaChar / (1.0 + math.exp(((EAccept + Vef) / KbT)))) + (NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -9.2e+120)
		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(Ev / KbT)))));
	elseif (Ev <= -3.7e+40)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Vef) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -9.2e+120)
		tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (Ev <= -3.7e+40)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = (NaChar / (1.0 + exp(((EAccept + Vef) / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -9.2e+120], 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[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, -3.7e+40], 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[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + Vef), $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]]]

Derivation

1. Split input into 3 regimes

2. if Ev < -9.1999999999999997e120

   1. Initial program 100.0%

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

   2. Taylor expanded in Ev around inf

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

      1. lower-/.f64 68.9

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

   4. Applied rewrites 68.9%

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

   if -9.1999999999999997e120 < Ev < -3.7e40

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if -3.7e40 < Ev

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around 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

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-+.f64 75.7

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

   7. Applied rewrites 75.7%

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

Alternative 4: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -1.3 \cdot 10^{+215}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}}\\ \mathbf{elif}\;Ev \leq -5.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;Ev \leq -3.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + Vef}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -1.3e+215)
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ EAccept Ev) KbT))))
    (/ NdChar (+ 1.0 (exp (/ (- EDonor Ec) KbT)))))
   (if (<= Ev -5.8e+143)
     (+
      (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
      (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
     (if (<= Ev -3.7e+40)
       (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))
       (+
        (/ NaChar (+ 1.0 (exp (/ (+ EAccept Vef) KbT))))
        (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) 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 <= -1.3e+215) {
		tmp = (NaChar / (1.0 + exp(((EAccept + Ev) / KbT)))) + (NdChar / (1.0 + exp(((EDonor - Ec) / KbT))));
	} else if (Ev <= -5.8e+143) {
		tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (Ev <= -3.7e+40) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + exp(((EAccept + Vef) / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / 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 <= (-1.3d+215)) then
        tmp = (nachar / (1.0d0 + exp(((eaccept + ev) / kbt)))) + (ndchar / (1.0d0 + exp(((edonor - ec) / kbt))))
    else if (ev <= (-5.8d+143)) then
        tmp = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else if (ev <= (-3.7d+40)) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    else
        tmp = (nachar / (1.0d0 + exp(((eaccept + vef) / kbt)))) + (ndchar / (1.0d0 + exp((((edonor + vef) - ec) / 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 <= -1.3e+215) {
		tmp = (NaChar / (1.0 + Math.exp(((EAccept + Ev) / KbT)))) + (NdChar / (1.0 + Math.exp(((EDonor - Ec) / KbT))));
	} else if (Ev <= -5.8e+143) {
		tmp = (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else if (Ev <= -3.7e+40) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((EAccept + Vef) / KbT)))) + (NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -1.3e+215:
		tmp = (NaChar / (1.0 + math.exp(((EAccept + Ev) / KbT)))) + (NdChar / (1.0 + math.exp(((EDonor - Ec) / KbT))))
	elif Ev <= -5.8e+143:
		tmp = (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	elif Ev <= -3.7e+40:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	else:
		tmp = (NaChar / (1.0 + math.exp(((EAccept + Vef) / KbT)))) + (NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -1.3e+215)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Ev) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Ec) / KbT)))));
	elseif (Ev <= -5.8e+143)
		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)))));
	elseif (Ev <= -3.7e+40)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Vef) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -1.3e+215)
		tmp = (NaChar / (1.0 + exp(((EAccept + Ev) / KbT)))) + (NdChar / (1.0 + exp(((EDonor - Ec) / KbT))));
	elseif (Ev <= -5.8e+143)
		tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	elseif (Ev <= -3.7e+40)
		tmp = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	else
		tmp = (NaChar / (1.0 + exp(((EAccept + Vef) / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -1.3e+215], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, -5.8e+143], 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], If[LessEqual[Ev, -3.7e+40], 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[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + Vef), $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]]]]

Derivation

1. Split input into 4 regimes

2. if Ev < -1.3e215

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around 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

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in Vef around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 68.8

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

   7. Applied rewrites 68.8%

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

   if -1.3e215 < Ev < -5.7999999999999996e143

   1. Initial program 100.0%

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

   2. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 68.8

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

   4. Applied rewrites 68.8%

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

   if -5.7999999999999996e143 < Ev < -3.7e40

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if -3.7e40 < Ev

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around 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

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-+.f64 75.7

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

   7. Applied rewrites 75.7%

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

Alternative 5: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{EAccept + Vef}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}}\\ \mathbf{if}\;Vef \leq -128:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ EAccept Vef) KbT))))
          (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) KbT)))))))
   (if (<= Vef -128.0)
     t_0
     (if (<= Vef 1.65e-66)
       (+
        (/ NaChar (+ 1.0 (exp (/ (+ EAccept 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 + Vef) / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT))));
	double tmp;
	if (Vef <= -128.0) {
		tmp = t_0;
	} else if (Vef <= 1.65e-66) {
		tmp = (NaChar / (1.0 + exp(((EAccept + 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) :: tmp
    t_0 = (nachar / (1.0d0 + exp(((eaccept + vef) / kbt)))) + (ndchar / (1.0d0 + exp((((edonor + vef) - ec) / kbt))))
    if (vef <= (-128.0d0)) then
        tmp = t_0
    else if (vef <= 1.65d-66) then
        tmp = (nachar / (1.0d0 + exp(((eaccept + 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 + Vef) / KbT)))) + (NdChar / (1.0 + Math.exp((((EDonor + Vef) - Ec) / KbT))));
	double tmp;
	if (Vef <= -128.0) {
		tmp = t_0;
	} else if (Vef <= 1.65e-66) {
		tmp = (NaChar / (1.0 + Math.exp(((EAccept + 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 + Vef) / KbT)))) + (NdChar / (1.0 + math.exp((((EDonor + Vef) - Ec) / KbT))))
	tmp = 0
	if Vef <= -128.0:
		tmp = t_0
	elif Vef <= 1.65e-66:
		tmp = (NaChar / (1.0 + math.exp(((EAccept + 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(Float64(EAccept + Vef) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)))))
	tmp = 0.0
	if (Vef <= -128.0)
		tmp = t_0;
	elseif (Vef <= 1.65e-66)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + 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 + Vef) / KbT)))) + (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT))));
	tmp = 0.0;
	if (Vef <= -128.0)
		tmp = t_0;
	elseif (Vef <= 1.65e-66)
		tmp = (NaChar / (1.0 + exp(((EAccept + 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[(N[(EAccept + Vef), $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]}, If[LessEqual[Vef, -128.0], t$95$0, If[LessEqual[Vef, 1.65e-66], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / 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 2 regimes

2. if Vef < -128 or 1.6499999999999999e-66 < Vef

   1. Initial program 100.0%

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

   2. 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

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in Ev around 0

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

      1. lower-+.f64 75.7

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

   7. Applied rewrites 75.7%

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

   if -128 < Vef < 1.6499999999999999e-66

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around 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

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in Vef around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 68.8

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

   7. Applied rewrites 68.8%

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

Alternative 6: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + mu\right) - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-284}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0}\\ \mathbf{elif}\;t\_1 \leq 10^{-97}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NdChar, \frac{NaChar}{t\_0 - -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -2e-309)
     (+
      (/ NdChar (+ 1.0 (exp (/ (- (+ Vef mu) Ec) KbT))))
      (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
     (if (<= t_1 5e-284)
       (/ NaChar (+ 1.0 t_0))
       (if (<= t_1 1e-97)
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ EAccept Ev) KbT))))
          (/ NdChar (+ 1.0 (exp (/ (- EDonor Ec) KbT)))))
         (fma 0.5 NdChar (/ NaChar (- t_0 -1.0))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-309) {
		tmp = (NdChar / (1.0 + exp((((Vef + mu) - Ec) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (t_1 <= 5e-284) {
		tmp = NaChar / (1.0 + t_0);
	} else if (t_1 <= 1e-97) {
		tmp = (NaChar / (1.0 + exp(((EAccept + Ev) / KbT)))) + (NdChar / (1.0 + exp(((EDonor - Ec) / KbT))));
	} else {
		tmp = fma(0.5, NdChar, (NaChar / (t_0 - -1.0)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-309)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + mu) - Ec) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (t_1 <= 5e-284)
		tmp = Float64(NaChar / Float64(1.0 + t_0));
	elseif (t_1 <= 1e-97)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Ev) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Ec) / KbT)))));
	else
		tmp = fma(0.5, NdChar, Float64(NaChar / Float64(t_0 - -1.0)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + 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-309], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-284], N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-97], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * NdChar + N[(NaChar / N[(t$95$0 - -1.0), $MachinePrecision]), $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))))) < -1.9999999999999988e-309

   1. Initial program 100.0%

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

   2. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in mu around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in EDonor around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 62.9

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

   10. Applied rewrites 62.9%

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

   if -1.9999999999999988e-309 < (+.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))))) < 4.99999999999999973e-284

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if 4.99999999999999973e-284 < (+.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))))) < 1.00000000000000004e-97

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around 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

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in Vef around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 68.8

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

   7. Applied rewrites 68.8%

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

   if 1.00000000000000004e-97 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. associate-/r/ N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.0%

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

Alternative 7: 72.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + mu\right) - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NdChar, \frac{NaChar}{t\_0 - -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -2e-309)
     (+
      (/ NdChar (+ 1.0 (exp (/ (- (+ Vef mu) Ec) KbT))))
      (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
     (if (<= t_1 5e-7)
       (/ NaChar (+ 1.0 t_0))
       (fma 0.5 NdChar (/ NaChar (- t_0 -1.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-309) {
		tmp = (NdChar / (1.0 + exp((((Vef + mu) - Ec) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (t_1 <= 5e-7) {
		tmp = NaChar / (1.0 + t_0);
	} else {
		tmp = fma(0.5, NdChar, (NaChar / (t_0 - -1.0)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-309)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + mu) - Ec) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (t_1 <= 5e-7)
		tmp = Float64(NaChar / Float64(1.0 + t_0));
	else
		tmp = fma(0.5, NdChar, Float64(NaChar / Float64(t_0 - -1.0)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + 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-309], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * NdChar + N[(NaChar / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.9999999999999988e-309

   1. Initial program 100.0%

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

   2. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in mu around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in EDonor around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 62.9

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

   10. Applied rewrites 62.9%

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

   if -1.9999999999999988e-309 < (+.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))))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if 4.99999999999999977e-7 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. associate-/r/ N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.0%

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

Alternative 8: 71.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(EDonor + mu\right) - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NdChar, \frac{NaChar}{t\_0 - -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -2e-309)
     (+
      (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor mu) Ec) KbT))))
      (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
     (if (<= t_1 5e-7)
       (/ NaChar (+ 1.0 t_0))
       (fma 0.5 NdChar (/ NaChar (- t_0 -1.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-309) {
		tmp = (NdChar / (1.0 + exp((((EDonor + mu) - Ec) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (t_1 <= 5e-7) {
		tmp = NaChar / (1.0 + t_0);
	} else {
		tmp = fma(0.5, NdChar, (NaChar / (t_0 - -1.0)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-309)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + mu) - Ec) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (t_1 <= 5e-7)
		tmp = Float64(NaChar / Float64(1.0 + t_0));
	else
		tmp = fma(0.5, NdChar, Float64(NaChar / Float64(t_0 - -1.0)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + 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-309], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor + mu), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * NdChar + N[(NaChar / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.9999999999999988e-309

   1. Initial program 100.0%

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

   2. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in Vef around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 63.9

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

   7. Applied rewrites 63.9%

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

   if -1.9999999999999988e-309 < (+.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))))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if 4.99999999999999977e-7 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. associate-/r/ N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.0%

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

Alternative 9: 71.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(EDonor + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NdChar, \frac{NaChar}{t\_0 - -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -2e-309)
     (+
      (/ NdChar (+ 1.0 (exp (/ (- (+ EDonor Vef) Ec) KbT))))
      (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
     (if (<= t_1 5e-7)
       (/ NaChar (+ 1.0 t_0))
       (fma 0.5 NdChar (/ NaChar (- t_0 -1.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -2e-309) {
		tmp = (NdChar / (1.0 + exp((((EDonor + Vef) - Ec) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (t_1 <= 5e-7) {
		tmp = NaChar / (1.0 + t_0);
	} else {
		tmp = fma(0.5, NdChar, (NaChar / (t_0 - -1.0)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_1 <= -2e-309)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor + Vef) - Ec) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (t_1 <= 5e-7)
		tmp = Float64(NaChar / Float64(1.0 + t_0));
	else
		tmp = fma(0.5, NdChar, Float64(NaChar / Float64(t_0 - -1.0)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + 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-309], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * NdChar + N[(NaChar / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -1.9999999999999988e-309

   1. Initial program 100.0%

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

   2. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in mu around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 63.8

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

   7. Applied rewrites 63.8%

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

   if -1.9999999999999988e-309 < (+.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))))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if 4.99999999999999977e-7 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. associate-/r/ N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.0%

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

Alternative 10: 69.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-295}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, NdChar, \frac{NaChar}{t\_0 - -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -5e-295)
     (+
      (/ NdChar (+ 1.0 (exp (/ (- Vef Ec) KbT))))
      (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
     (if (<= t_1 5e-7)
       (/ NaChar (+ 1.0 t_0))
       (fma 0.5 NdChar (/ NaChar (- t_0 -1.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_1 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double tmp;
	if (t_1 <= -5e-295) {
		tmp = (NdChar / (1.0 + exp(((Vef - Ec) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (t_1 <= 5e-7) {
		tmp = NaChar / (1.0 + t_0);
	} else {
		tmp = fma(0.5, NdChar, (NaChar / (t_0 - -1.0)));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	tmp = 0.0
	if (t_1 <= -5e-295)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Vef - Ec) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (t_1 <= 5e-7)
		tmp = Float64(NaChar / Float64(1.0 + t_0));
	else
		tmp = fma(0.5, NdChar, Float64(NaChar / Float64(t_0 - -1.0)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-295], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(Vef - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * NdChar + N[(NaChar / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000008e-295

   1. Initial program 100.0%

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

   2. Taylor expanded in EAccept around inf

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

      1. lower-/.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in mu around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in EDonor around 0

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

      1. lower--.f64 57.3

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

   10. Applied rewrites 57.3%

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

   if -5.00000000000000008e-295 < (+.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))))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if 4.99999999999999977e-7 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. associate-/r/ N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.0%

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

Alternative 11: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}\\ t_1 := t\_0 + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ t_2 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_3 := \mathsf{fma}\left(0.5, NdChar, \frac{NaChar}{t\_2 - -1}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+144}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-232}:\\ \;\;\;\;t\_0 + 0.5 \cdot NaChar\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))))
        (t_1
         (+
          t_0
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT))))))
        (t_2 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_3 (fma 0.5 NdChar (/ NaChar (- t_2 -1.0)))))
   (if (<= t_1 -2e+144)
     t_3
     (if (<= t_1 -1e-232)
       (+ t_0 (* 0.5 NaChar))
       (if (<= t_1 5e-7) (/ NaChar (+ 1.0 t_2)) t_3)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)));
	double t_1 = t_0 + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double t_2 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_3 = fma(0.5, NdChar, (NaChar / (t_2 - -1.0)));
	double tmp;
	if (t_1 <= -2e+144) {
		tmp = t_3;
	} else if (t_1 <= -1e-232) {
		tmp = t_0 + (0.5 * NaChar);
	} else if (t_1 <= 5e-7) {
		tmp = NaChar / (1.0 + t_2);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT))))
	t_1 = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	t_2 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_3 = fma(0.5, NdChar, Float64(NaChar / Float64(t_2 - -1.0)))
	tmp = 0.0
	if (t_1 <= -2e+144)
		tmp = t_3;
	elseif (t_1 <= -1e-232)
		tmp = Float64(t_0 + Float64(0.5 * NaChar));
	elseif (t_1 <= 5e-7)
		tmp = Float64(NaChar / Float64(1.0 + t_2));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * NdChar + N[(NaChar / N[(t$95$2 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+144], t$95$3, If[LessEqual[t$95$1, -1e-232], N[(t$95$0 + N[(0.5 * NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(NaChar / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

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))))) < -2.00000000000000005e144 or 4.99999999999999977e-7 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. associate-/r/ N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.0%

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

   if -2.00000000000000005e144 < (+.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))))) < -1.00000000000000002e-232

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-*.f64 46.6

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

   4. Applied rewrites 46.6%

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

   if -1.00000000000000002e-232 < (+.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))))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

Alternative 12: 67.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ t_1 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_2 := \mathsf{fma}\left(0.5, NdChar, \frac{NaChar}{t\_1 - -1}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-216}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT))))))
        (t_1 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_2 (fma 0.5 NdChar (/ NaChar (- t_1 -1.0)))))
   (if (<= t_0 -5e-216) t_2 (if (<= t_0 5e-7) (/ NaChar (+ 1.0 t_1)) t_2))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
	double t_1 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_2 = fma(0.5, NdChar, (NaChar / (t_1 - -1.0)));
	double tmp;
	if (t_0 <= -5e-216) {
		tmp = t_2;
	} else if (t_0 <= 5e-7) {
		tmp = NaChar / (1.0 + t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
	t_1 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_2 = fma(0.5, NdChar, Float64(NaChar / Float64(t_1 - -1.0)))
	tmp = 0.0
	if (t_0 <= -5e-216)
		tmp = t_2;
	elseif (t_0 <= 5e-7)
		tmp = Float64(NaChar / Float64(1.0 + t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * NdChar + N[(NaChar / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-216], t$95$2, If[LessEqual[t$95$0, 5e-7], N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000021e-216 or 4.99999999999999977e-7 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. associate-/r/ N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in KbT around inf

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

   6. Applied rewrites 47.0%

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

   if -5.00000000000000021e-216 < (+.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))))) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

Alternative 13: 64.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.5 \cdot 10^{+172}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 6.2 \cdot 10^{+211}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.5e+172)
     t_0
     (if (<= KbT 6.2e+211)
       (/ 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.5e+172) {
		tmp = t_0;
	} else if (KbT <= 6.2e+211) {
		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) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.5d+172)) then
        tmp = t_0
    else if (kbt <= 6.2d+211) 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.5e+172) {
		tmp = t_0;
	} else if (KbT <= 6.2e+211) {
		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 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.5e+172:
		tmp = t_0
	elif KbT <= 6.2e+211:
		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(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.5e+172)
		tmp = t_0;
	elseif (KbT <= 6.2e+211)
		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 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.5e+172)
		tmp = t_0;
	elseif (KbT <= 6.2e+211)
		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[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.5e+172], t$95$0, If[LessEqual[KbT, 6.2e+211], 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 KbT < -1.5e172 or 6.2000000000000003e211 < KbT

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.5

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

   4. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 27.5

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

   6. Applied rewrites 27.5%

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

   if -1.5e172 < KbT < 6.2000000000000003e211

   1. Initial program 100.0%

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

   2. Taylor expanded in NdChar around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

Alternative 14: 58.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.25 \cdot 10^{+172}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.5 \cdot 10^{+201}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept + \left(Ev + Vef\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.25e+172)
     t_0
     (if (<= KbT 1.5e+201)
       (/ NaChar (+ 1.0 (exp (/ (+ EAccept (+ Ev Vef)) KbT))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.25e+172) {
		tmp = t_0;
	} else if (KbT <= 1.5e+201) {
		tmp = NaChar / (1.0 + exp(((EAccept + (Ev + Vef)) / 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 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.25d+172)) then
        tmp = t_0
    else if (kbt <= 1.5d+201) then
        tmp = nachar / (1.0d0 + exp(((eaccept + (ev + vef)) / 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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.25e+172) {
		tmp = t_0;
	} else if (KbT <= 1.5e+201) {
		tmp = NaChar / (1.0 + Math.exp(((EAccept + (Ev + Vef)) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.25e+172:
		tmp = t_0
	elif KbT <= 1.5e+201:
		tmp = NaChar / (1.0 + math.exp(((EAccept + (Ev + Vef)) / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.25e+172)
		tmp = t_0;
	elseif (KbT <= 1.5e+201)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(EAccept + Float64(Ev + Vef)) / 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 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.25e+172)
		tmp = t_0;
	elseif (KbT <= 1.5e+201)
		tmp = NaChar / (1.0 + exp(((EAccept + (Ev + Vef)) / 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[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.25e+172], t$95$0, If[LessEqual[KbT, 1.5e+201], N[(NaChar / N[(1.0 + N[Exp[N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -1.25e172 or 1.50000000000000012e201 < KbT

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.5

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

   4. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 27.5

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

   6. Applied rewrites 27.5%

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

   if -1.25e172 < KbT < 1.50000000000000012e201

   1. Initial program 100.0%

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

   2. Taylor expanded in mu around 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

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

   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. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 54.9

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

   7. Applied rewrites 54.9%

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

Alternative 15: 31.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-168}:\\ \;\;\;\;\frac{1}{\frac{2 + -2 \cdot \frac{NdChar}{NaChar}}{NaChar}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar 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-232)
     t_0
     (if (<= t_1 1e-168)
       (/ 1.0 (/ (+ 2.0 (* -2.0 (/ NdChar NaChar))) NaChar))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double 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-232) {
		tmp = t_0;
	} else if (t_1 <= 1e-168) {
		tmp = 1.0 / ((2.0 + (-2.0 * (NdChar / NaChar))) / NaChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (t_1 <= (-1d-232)) then
        tmp = t_0
    else if (t_1 <= 1d-168) then
        tmp = 1.0d0 / ((2.0d0 + ((-2.0d0) * (ndchar / nachar))) / nachar)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	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-232) {
		tmp = t_0;
	} else if (t_1 <= 1e-168) {
		tmp = 1.0 / ((2.0 + (-2.0 * (NdChar / NaChar))) / NaChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + 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-232:
		tmp = t_0
	elif t_1 <= 1e-168:
		tmp = 1.0 / ((2.0 + (-2.0 * (NdChar / NaChar))) / NaChar)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + 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-232)
		tmp = t_0;
	elseif (t_1 <= 1e-168)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(-2.0 * Float64(NdChar / NaChar))) / NaChar));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	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-232)
		tmp = t_0;
	elseif (t_1 <= 1e-168)
		tmp = 1.0 / ((2.0 + (-2.0 * (NdChar / NaChar))) / NaChar);
	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[(0.5 * N[(NdChar + 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-232], t$95$0, If[LessEqual[t$95$1, 1e-168], N[(1.0 / N[(N[(2.0 + N[(-2.0 * N[(NdChar / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / NaChar), $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))))) < -1.00000000000000002e-232 or 1e-168 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.5

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

   4. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 27.5

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

   6. Applied rewrites 27.5%

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

   if -1.00000000000000002e-232 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 1e-168

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.5

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

   4. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. div-flip N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{2} \cdot \left(NaChar - NdChar\right)}{\mathsf{fma}\left(\frac{1}{2}, NaChar, \frac{1}{2} \cdot NdChar\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot NaChar} - \frac{1}{2} \cdot NdChar\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{2} \cdot \left(NaChar - NdChar\right)}{\mathsf{fma}\left(\frac{1}{2}, NaChar, \frac{1}{2} \cdot NdChar\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot NaChar - \frac{1}{2} \cdot NdChar\right)}}} \]

      13. lift-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-out-- N/A

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

      21. lower-*.f64 N/A

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

      22. lower--.f64 17.4

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

   6. Applied rewrites 17.4%

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

   7. Taylor expanded in NaChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 19.9

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

   9. Applied rewrites 19.9%

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

Alternative 16: 31.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ t_1 := \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{\frac{2 + -2 \cdot \frac{NaChar}{NdChar}}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar)))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT)))))))
   (if (<= t_1 -2e-219)
     t_0
     (if (<= t_1 4e-259)
       (/ 1.0 (/ (+ 2.0 (* -2.0 (/ NaChar NdChar))) NdChar))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + 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 <= -2e-219) {
		tmp = t_0;
	} else if (t_1 <= 4e-259) {
		tmp = 1.0 / ((2.0 + (-2.0 * (NaChar / NdChar))) / NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
use fmin_fmax_functions
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    t_1 = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
    if (t_1 <= (-2d-219)) then
        tmp = t_0
    else if (t_1 <= 4d-259) then
        tmp = 1.0d0 / ((2.0d0 + ((-2.0d0) * (nachar / ndchar))) / ndchar)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + 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 <= -2e-219) {
		tmp = t_0;
	} else if (t_1 <= 4e-259) {
		tmp = 1.0 / ((2.0 + (-2.0 * (NaChar / NdChar))) / NdChar);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + 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 <= -2e-219:
		tmp = t_0
	elif t_1 <= 4e-259:
		tmp = 1.0 / ((2.0 + (-2.0 * (NaChar / NdChar))) / NdChar)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + 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 <= -2e-219)
		tmp = t_0;
	elseif (t_1 <= 4e-259)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(-2.0 * Float64(NaChar / NdChar))) / NdChar));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + 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 <= -2e-219)
		tmp = t_0;
	elseif (t_1 <= 4e-259)
		tmp = 1.0 / ((2.0 + (-2.0 * (NaChar / NdChar))) / NdChar);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + 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, -2e-219], t$95$0, If[LessEqual[t$95$1, 4e-259], N[(1.0 / N[(N[(2.0 + N[(-2.0 * N[(NaChar / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.0000000000000001e-219 or 4.0000000000000003e-259 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.5

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

   4. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 27.5

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

   6. Applied rewrites 27.5%

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

   if -2.0000000000000001e-219 < (+.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))))) < 4.0000000000000003e-259

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.5

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

   4. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. div-flip N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{2} \cdot \left(NaChar - NdChar\right)}{\mathsf{fma}\left(\frac{1}{2}, NaChar, \frac{1}{2} \cdot NdChar\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot NaChar} - \frac{1}{2} \cdot NdChar\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{2} \cdot \left(NaChar - NdChar\right)}{\mathsf{fma}\left(\frac{1}{2}, NaChar, \frac{1}{2} \cdot NdChar\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot NaChar - \frac{1}{2} \cdot NdChar\right)}}} \]

      13. lift-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-out-- N/A

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

      21. lower-*.f64 N/A

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

      22. lower--.f64 17.4

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

   6. Applied rewrites 17.4%

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

   7. Taylor expanded in NdChar around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 19.2

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

   9. Applied rewrites 19.2%

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

Alternative 17: 27.5% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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 * (ndchar + nachar)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. 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 27.5

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

4. Applied rewrites 27.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, NaChar, 0.5 \cdot NdChar\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 27.5

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

6. Applied rewrites 27.5%

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

Alternative 18: 22.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -7.5 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{elif}\;NaChar \leq 2.05 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -7.5e-64)
   (* 0.5 NaChar)
   (if (<= NaChar 2.05e+129) (* 0.5 NdChar) (* 0.5 NaChar))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -7.5e-64) {
		tmp = 0.5 * NaChar;
	} else if (NaChar <= 2.05e+129) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = 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 (nachar <= (-7.5d-64)) then
        tmp = 0.5d0 * nachar
    else if (nachar <= 2.05d+129) then
        tmp = 0.5d0 * ndchar
    else
        tmp = 0.5d0 * nachar
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -7.5e-64) {
		tmp = 0.5 * NaChar;
	} else if (NaChar <= 2.05e+129) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = 0.5 * NaChar;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -7.5e-64:
		tmp = 0.5 * NaChar
	elif NaChar <= 2.05e+129:
		tmp = 0.5 * NdChar
	else:
		tmp = 0.5 * NaChar
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -7.5e-64)
		tmp = Float64(0.5 * NaChar);
	elseif (NaChar <= 2.05e+129)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = Float64(0.5 * NaChar);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -7.5e-64)
		tmp = 0.5 * NaChar;
	elseif (NaChar <= 2.05e+129)
		tmp = 0.5 * NdChar;
	else
		tmp = 0.5 * NaChar;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -7.5e-64], N[(0.5 * NaChar), $MachinePrecision], If[LessEqual[NaChar, 2.05e+129], N[(0.5 * NdChar), $MachinePrecision], N[(0.5 * NaChar), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -7.49999999999999949e-64 or 2.0500000000000001e129 < NaChar

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.5

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

   4. Applied rewrites 27.5%

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

   5. Taylor expanded in NdChar around 0

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

      1. lower-*.f64 18.6

         \[\leadsto 0.5 \cdot NaChar \]

   7. Applied rewrites 18.6%

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

   if -7.49999999999999949e-64 < NaChar < 2.0500000000000001e129

   1. Initial program 100.0%

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

   2. Taylor expanded in KbT around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 27.5

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

   4. Applied rewrites 27.5%

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

   5. Taylor expanded in NdChar around 0

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

      1. lower-*.f64 18.6

         \[\leadsto 0.5 \cdot NaChar \]

   7. Applied rewrites 18.6%

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

   8. Taylor expanded in NdChar around inf

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

      1. lower-*.f64 17.6

         \[\leadsto 0.5 \cdot NdChar \]

   10. Applied rewrites 17.6%

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

Alternative 19: 18.6% 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 100.0%

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

2. Taylor expanded in KbT around inf

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 27.5

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

4. Applied rewrites 27.5%

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

5. Taylor expanded in NdChar around 0

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

   1. lower-*.f64 18.6

      \[\leadsto 0.5 \cdot NaChar \]

7. Applied rewrites 18.6%

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

Reproduce

herbie shell --seed 2025156 
(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))))))