Jmat.Real.lambertw, newton loop step

Percentage Accurate: 76.6% → 99.3%

Time: 16.4s

Alternatives: 22

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.6047624280250576e+249
  2. x = -5.134921040290169e-55

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t_0 - x}{e^{wj} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))) (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	return wj - ((t_0 - x) / (exp(wj) + t_0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = wj * exp(wj)
    code = wj - ((t_0 - x) / (exp(wj) + t_0))
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	return wj - ((t_0 - x) / (Math.exp(wj) + t_0));
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	return wj - ((t_0 - x) / (math.exp(wj) + t_0))

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	return Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
end

MATLAB

function tmp = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = wj - ((t_0 - x) / (exp(wj) + t_0));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t_0 - x}{e^{wj} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))) (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	return wj - ((t_0 - x) / (exp(wj) + t_0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = wj * exp(wj)
    code = wj - ((t_0 - x) / (exp(wj) + t_0))
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	return wj - ((t_0 - x) / (Math.exp(wj) + t_0));
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	return wj - ((t_0 - x) / (math.exp(wj) + t_0))

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	return Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
end

MATLAB

function tmp = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = wj - ((t_0 - x) / (exp(wj) + t_0));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \mathbf{elif}\;wj \leq 3 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{\frac{1}{wj}}{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.2e-6)
   (+ wj (* (- wj (/ x (exp wj))) (/ -1.0 (+ wj 1.0))))
   (if (<= wj 3e-6)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (- (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5)))) (pow wj 3.0))))
     (/ 1.0 (+ (/ 1.0 wj) (/ (/ 1.0 wj) wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.2e-6) {
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (wj + 1.0)));
	} else if (wj <= 3e-6) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - pow(wj, 3.0)));
	} else {
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-1.2d-6)) then
        tmp = wj + ((wj - (x / exp(wj))) * ((-1.0d0) / (wj + 1.0d0)))
    else if (wj <= 3d-6) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))) - (wj ** 3.0d0)))
    else
        tmp = 1.0d0 / ((1.0d0 / wj) + ((1.0d0 / wj) / wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1.2e-6) {
		tmp = wj + ((wj - (x / Math.exp(wj))) * (-1.0 / (wj + 1.0)));
	} else if (wj <= 3e-6) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - Math.pow(wj, 3.0)));
	} else {
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.2e-6:
		tmp = wj + ((wj - (x / math.exp(wj))) * (-1.0 / (wj + 1.0)))
	elif wj <= 3e-6:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - math.pow(wj, 3.0)))
	else:
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.2e-6)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) * Float64(-1.0 / Float64(wj + 1.0))));
	elseif (wj <= 3e-6)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - (wj ^ 3.0))));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / wj) + Float64(Float64(1.0 / wj) / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.2e-6)
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (wj + 1.0)));
	elseif (wj <= 3e-6)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - (wj ^ 3.0)));
	else
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.2e-6], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3e-6], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / wj), $MachinePrecision] + N[(N[(1.0 / wj), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.1999999999999999e-6

   1. Initial program 84.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 94.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 94.5%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 84.5%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 84.5%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 94.5%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 94.5%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 93.8%

         \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \frac{x}{e^{wj}}}}} \]

      2. associate-/r/ 94.6%

         \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)} \]

   6. Applied egg-rr 94.6%

      \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)} \]

   if -1.1999999999999999e-6 < wj < 3.0000000000000001e-6

   1. Initial program 76.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 76.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 76.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 76.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 76.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 76.7%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 76.7%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \color{blue}{{wj}^{3}} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \]

   if 3.0000000000000001e-6 < wj

   1. Initial program 23.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 23.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 23.9%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 95.5%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 95.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 95.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 95.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in wj around inf 99.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{wj} + \frac{1}{{wj}^{2}}}} \]
   8. Step-by-step derivation

      1. inv-pow 99.3%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{\left({wj}^{2}\right)}^{-1}}} \]

      2. unpow2 99.3%

         \[\leadsto \frac{1}{\frac{1}{wj} + {\color{blue}{\left(wj \cdot wj\right)}}^{-1}} \]

      3. pow-prod-down 99.0%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{wj}^{-1} \cdot {wj}^{-1}}} \]

      4. inv-pow 99.0%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj}} \cdot {wj}^{-1}} \]

      5. inv-pow 99.0%

         \[\leadsto \frac{1}{\frac{1}{wj} + \frac{1}{wj} \cdot \color{blue}{\frac{1}{wj}}} \]

   9. Applied egg-rr 99.0%

      \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj} \cdot \frac{1}{wj}}} \]
   10. Step-by-step derivation

       1. un-div-inv 99.3%

          \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{\frac{1}{wj}}{wj}}} \]

   11. Applied egg-rr 99.3%

       \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{\frac{1}{wj}}{wj}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \mathbf{elif}\;wj \leq 3 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{\frac{1}{wj}}{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x \cdot -4 + x \cdot 1.5\\ t_2 := \frac{x}{e^{wj}}\\ t_3 := wj - t_2\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_1 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{t_2 - wj}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, t_3, \frac{t_3}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))
        (t_1 (+ (* x -4.0) (* x 1.5)))
        (t_2 (/ x (exp wj)))
        (t_3 (- wj t_2)))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-18)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (+
        (*
         (pow wj 3.0)
         (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_1) (* x 0.6666666666666666)))))
        (* (pow wj 2.0) (- 1.0 t_1)))))
     (+
      (+ wj (/ (- t_2 wj) (+ wj 1.0)))
      (fma (/ -1.0 (+ wj 1.0)) t_3 (/ t_3 (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double t_2 = x / exp(wj);
	double t_3 = wj - t_2;
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-18) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_1))));
	} else {
		tmp = (wj + ((t_2 - wj) / (wj + 1.0))) + fma((-1.0 / (wj + 1.0)), t_3, (t_3 / (wj + 1.0)));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	t_2 = Float64(x / exp(wj))
	t_3 = Float64(wj - t_2)
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 5e-18)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 3.0) * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_1) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_1)))));
	else
		tmp = Float64(Float64(wj + Float64(Float64(t_2 - wj) / Float64(wj + 1.0))) + fma(Float64(-1.0 / Float64(wj + 1.0)), t_3, Float64(t_3 / Float64(wj + 1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(wj - t$95$2), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-18], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$1), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj + N[(N[(t$95$2 - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$3 + N[(t$95$3 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5.00000000000000036e-18

   1. Initial program 69.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 70.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 70.5%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 70.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 70.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 70.5%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 70.5%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 70.5%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.1%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)} \]

   if 5.00000000000000036e-18 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 91.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 91.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 91.2%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 91.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 91.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 98.7%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 98.7%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-un-lft-identity 98.7%

         \[\leadsto \color{blue}{1 \cdot wj} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \]

      2. div-inv 98.7%

         \[\leadsto 1 \cdot wj - \color{blue}{\left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}} \]

      3. prod-diff 98.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, wj, -\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right) + \mathsf{fma}\left(-\frac{1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right)} \]

      4. associate-/r/ 98.4%

         \[\leadsto \mathsf{fma}\left(1, wj, -\color{blue}{\frac{1}{\frac{wj + 1}{wj - \frac{x}{e^{wj}}}}}\right) + \mathsf{fma}\left(-\frac{1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right) \]

      5. clear-num 98.7%

         \[\leadsto \mathsf{fma}\left(1, wj, -\color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\right) + \mathsf{fma}\left(-\frac{1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right) \]

      6. fma-neg 98.7%

         \[\leadsto \color{blue}{\left(1 \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)} + \mathsf{fma}\left(-\frac{1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right) \]

      7. *-un-lft-identity 98.7%

         \[\leadsto \left(\color{blue}{wj} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \mathsf{fma}\left(-\frac{1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right) \]

      8. associate-/r/ 98.4%

         \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \mathsf{fma}\left(-\frac{1}{wj + 1}, wj - \frac{x}{e^{wj}}, \color{blue}{\frac{1}{\frac{wj + 1}{wj - \frac{x}{e^{wj}}}}}\right) \]

      9. clear-num 98.8%

         \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \mathsf{fma}\left(-\frac{1}{wj + 1}, wj - \frac{x}{e^{wj}}, \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\right) \]

   6. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \mathsf{fma}\left(-\frac{1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)} \]
   7. Step-by-step derivation

      1. distribute-neg-frac 98.8%

         \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \mathsf{fma}\left(\color{blue}{\frac{-1}{wj + 1}}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) \]

      2. metadata-eval 98.8%

         \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \mathsf{fma}\left(\frac{\color{blue}{-1}}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) \]

   8. Simplified 98.8%

      \[\leadsto \color{blue}{\left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 10^{-11}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_1 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ (* x -4.0) (* x 1.5))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1e-11)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (+
        (*
         (pow wj 3.0)
         (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_1) (* x 0.6666666666666666)))))
        (* (pow wj 2.0) (- 1.0 t_1)))))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1e-11) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_1))));
	} else {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = (x * (-4.0d0)) + (x * 1.5d0)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1d-11) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_1) + (x * 0.6666666666666666d0))))) + ((wj ** 2.0d0) * (1.0d0 - t_1))))
    else
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 1e-11) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (Math.pow(wj, 2.0) * (1.0 - t_1))));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 1e-11:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (math.pow(wj, 2.0) * (1.0 - t_1))))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 1e-11)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 3.0) * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_1) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_1)))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1e-11)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + ((wj ^ 2.0) * (1.0 - t_1))));
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-11], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$1), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 9.99999999999999939e-12

   1. Initial program 70.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 70.6%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 70.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 70.1%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 70.1%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 70.7%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 70.7%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.0%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)} \]

   if 9.99999999999999939e-12 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 91.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 91.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 91.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 91.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 91.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 99.0%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 99.0%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-11}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1 \cdot 10^{-8}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + \left({wj}^{2} - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1e-8)
   (+ wj (* (- wj (/ x (exp wj))) (/ -1.0 (+ wj 1.0))))
   (if (<= wj 1.3e-8)
     (+ x (- (pow wj 2.0) (pow wj 3.0)))
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1e-8) {
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (wj + 1.0)));
	} else if (wj <= 1.3e-8) {
		tmp = x + (pow(wj, 2.0) - pow(wj, 3.0));
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-1d-8)) then
        tmp = wj + ((wj - (x / exp(wj))) * ((-1.0d0) / (wj + 1.0d0)))
    else if (wj <= 1.3d-8) then
        tmp = x + ((wj ** 2.0d0) - (wj ** 3.0d0))
    else
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1e-8) {
		tmp = wj + ((wj - (x / Math.exp(wj))) * (-1.0 / (wj + 1.0)));
	} else if (wj <= 1.3e-8) {
		tmp = x + (Math.pow(wj, 2.0) - Math.pow(wj, 3.0));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1e-8:
		tmp = wj + ((wj - (x / math.exp(wj))) * (-1.0 / (wj + 1.0)))
	elif wj <= 1.3e-8:
		tmp = x + (math.pow(wj, 2.0) - math.pow(wj, 3.0))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1e-8)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) * Float64(-1.0 / Float64(wj + 1.0))));
	elseif (wj <= 1.3e-8)
		tmp = Float64(x + Float64((wj ^ 2.0) - (wj ^ 3.0)));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1e-8)
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (wj + 1.0)));
	elseif (wj <= 1.3e-8)
		tmp = x + ((wj ^ 2.0) - (wj ^ 3.0));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1e-8], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.3e-8], N[(x + N[(N[Power[wj, 2.0], $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1e-8

   1. Initial program 86.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 95.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 95.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 87.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 87.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 94.7%

         \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \frac{x}{e^{wj}}}}} \]

      2. associate-/r/ 95.4%

         \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)} \]

   6. Applied egg-rr 95.4%

      \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)} \]

   if -1e-8 < wj < 1.3000000000000001e-8

   1. Initial program 76.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 76.2%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 76.2%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 76.2%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 76.2%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 76.2%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \color{blue}{{wj}^{3}} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \]

   7. Taylor expanded in x around 0 99.7%

      \[\leadsto x + \color{blue}{\left(-1 \cdot {wj}^{3} + {wj}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto x + \color{blue}{\left({wj}^{2} + -1 \cdot {wj}^{3}\right)} \]

      2. mul-1-neg 99.7%

         \[\leadsto x + \left({wj}^{2} + \color{blue}{\left(-{wj}^{3}\right)}\right) \]

      3. unsub-neg 99.7%

         \[\leadsto x + \color{blue}{\left({wj}^{2} - {wj}^{3}\right)} \]

   9. Simplified 99.7%

      \[\leadsto x + \color{blue}{\left({wj}^{2} - {wj}^{3}\right)} \]

   if 1.3000000000000001e-8 < wj

   1. Initial program 46.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 46.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 46.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 46.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 46.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 96.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 96.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 96.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 96.6%

         \[\leadsto wj - \frac{wj - \color{blue}{\frac{1}{\frac{e^{wj}}{x}}}}{wj + 1} \]

      2. associate-/r/ 96.4%

         \[\leadsto wj - \frac{wj - \color{blue}{\frac{1}{e^{wj}} \cdot x}}{wj + 1} \]

      3. rec-exp 96.6%

         \[\leadsto wj - \frac{wj - \color{blue}{e^{-wj}} \cdot x}{wj + 1} \]

   6. Applied egg-rr 96.6%

      \[\leadsto wj - \frac{wj - \color{blue}{e^{-wj} \cdot x}}{wj + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1 \cdot 10^{-8}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + \left({wj}^{2} - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.8 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.8e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (if (<= wj 3.7e-9)
     (+ x (+ (* -2.0 (* wj x)) (pow wj 2.0)))
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5.8e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 3.7e-9) {
		tmp = x + ((-2.0 * (wj * x)) + pow(wj, 2.0));
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-5.8d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else if (wj <= 3.7d-9) then
        tmp = x + (((-2.0d0) * (wj * x)) + (wj ** 2.0d0))
    else
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -5.8e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 3.7e-9) {
		tmp = x + ((-2.0 * (wj * x)) + Math.pow(wj, 2.0));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -5.8e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	elif wj <= 3.7e-9:
		tmp = x + ((-2.0 * (wj * x)) + math.pow(wj, 2.0))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5.8e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 3.7e-9)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + (wj ^ 2.0)));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -5.8e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	elseif (wj <= 3.7e-9)
		tmp = x + ((-2.0 * (wj * x)) + (wj ^ 2.0));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5.8e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.7e-9], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -5.79999999999999982e-9

   1. Initial program 86.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 95.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 95.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 87.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 87.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   if -5.79999999999999982e-9 < wj < 3.7e-9

   1. Initial program 76.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 76.2%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 76.2%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 76.2%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 76.2%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 76.2%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.3%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]

   6. Taylor expanded in x around 0 99.3%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \color{blue}{{wj}^{2}}\right) \]

   if 3.7e-9 < wj

   1. Initial program 46.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 46.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 46.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 46.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 46.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 96.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 96.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 96.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 96.6%

         \[\leadsto wj - \frac{wj - \color{blue}{\frac{1}{\frac{e^{wj}}{x}}}}{wj + 1} \]

      2. associate-/r/ 96.4%

         \[\leadsto wj - \frac{wj - \color{blue}{\frac{1}{e^{wj}} \cdot x}}{wj + 1} \]

      3. rec-exp 96.6%

         \[\leadsto wj - \frac{wj - \color{blue}{e^{-wj}} \cdot x}{wj + 1} \]

   6. Applied egg-rr 96.6%

      \[\leadsto wj - \frac{wj - \color{blue}{e^{-wj} \cdot x}}{wj + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.8 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.85 \cdot 10^{-8}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(\left(1 + x \cdot 2.5\right) - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.85e-8)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (if (<= wj 5.2e-9)
     (+ x (* (pow wj 2.0) (- (+ 1.0 (* x 2.5)) wj)))
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.85e-8) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 5.2e-9) {
		tmp = x + (pow(wj, 2.0) * ((1.0 + (x * 2.5)) - wj));
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-1.85d-8)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else if (wj <= 5.2d-9) then
        tmp = x + ((wj ** 2.0d0) * ((1.0d0 + (x * 2.5d0)) - wj))
    else
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1.85e-8) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 5.2e-9) {
		tmp = x + (Math.pow(wj, 2.0) * ((1.0 + (x * 2.5)) - wj));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.85e-8:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	elif wj <= 5.2e-9:
		tmp = x + (math.pow(wj, 2.0) * ((1.0 + (x * 2.5)) - wj))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.85e-8)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 5.2e-9)
		tmp = Float64(x + Float64((wj ^ 2.0) * Float64(Float64(1.0 + Float64(x * 2.5)) - wj)));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.85e-8)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	elseif (wj <= 5.2e-9)
		tmp = x + ((wj ^ 2.0) * ((1.0 + (x * 2.5)) - wj));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.85e-8], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5.2e-9], N[(x + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.85e-8

   1. Initial program 86.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 95.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 95.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 87.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 87.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   if -1.85e-8 < wj < 5.2000000000000002e-9

   1. Initial program 76.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 76.2%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 76.2%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 76.2%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 76.2%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 76.2%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \color{blue}{{wj}^{3}} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \]

   7. Taylor expanded in wj around inf 99.7%

      \[\leadsto x + \color{blue}{\left(-1 \cdot {wj}^{3} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.7%

         \[\leadsto x + \left(\color{blue}{\left(-{wj}^{3}\right)} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      2. cube-mult 99.7%

         \[\leadsto x + \left(\left(-\color{blue}{wj \cdot \left(wj \cdot wj\right)}\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      3. unpow2 99.7%

         \[\leadsto x + \left(\left(-wj \cdot \color{blue}{{wj}^{2}}\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      4. distribute-lft-neg-out 99.7%

         \[\leadsto x + \left(\color{blue}{\left(-wj\right) \cdot {wj}^{2}} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      5. *-commutative 99.7%

         \[\leadsto x + \left(\left(-wj\right) \cdot {wj}^{2} + \color{blue}{\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}}\right) \]

      6. distribute-rgt-out 99.7%

         \[\leadsto x + \left(\left(-wj\right) \cdot {wj}^{2} + \left(1 - \color{blue}{x \cdot \left(-4 + 1.5\right)}\right) \cdot {wj}^{2}\right) \]

      7. metadata-eval 99.7%

         \[\leadsto x + \left(\left(-wj\right) \cdot {wj}^{2} + \left(1 - x \cdot \color{blue}{-2.5}\right) \cdot {wj}^{2}\right) \]

      8. distribute-rgt-out 99.7%

         \[\leadsto x + \color{blue}{{wj}^{2} \cdot \left(\left(-wj\right) + \left(1 - x \cdot -2.5\right)\right)} \]

      9. sub-neg 99.7%

         \[\leadsto x + {wj}^{2} \cdot \left(\left(-wj\right) + \color{blue}{\left(1 + \left(-x \cdot -2.5\right)\right)}\right) \]

      10. distribute-rgt-neg-in 99.7%

          \[\leadsto x + {wj}^{2} \cdot \left(\left(-wj\right) + \left(1 + \color{blue}{x \cdot \left(--2.5\right)}\right)\right) \]

      11. metadata-eval 99.7%

          \[\leadsto x + {wj}^{2} \cdot \left(\left(-wj\right) + \left(1 + x \cdot \color{blue}{2.5}\right)\right) \]

   9. Simplified 99.7%

      \[\leadsto x + \color{blue}{{wj}^{2} \cdot \left(\left(-wj\right) + \left(1 + x \cdot 2.5\right)\right)} \]

   if 5.2000000000000002e-9 < wj

   1. Initial program 46.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 46.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 46.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 46.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 46.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 96.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 96.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 96.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 96.6%

         \[\leadsto wj - \frac{wj - \color{blue}{\frac{1}{\frac{e^{wj}}{x}}}}{wj + 1} \]

      2. associate-/r/ 96.4%

         \[\leadsto wj - \frac{wj - \color{blue}{\frac{1}{e^{wj}} \cdot x}}{wj + 1} \]

      3. rec-exp 96.6%

         \[\leadsto wj - \frac{wj - \color{blue}{e^{-wj}} \cdot x}{wj + 1} \]

   6. Applied egg-rr 96.6%

      \[\leadsto wj - \frac{wj - \color{blue}{e^{-wj} \cdot x}}{wj + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.85 \cdot 10^{-8}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(\left(1 + x \cdot 2.5\right) - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6.8 \cdot 10^{-9}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \mathbf{elif}\;wj \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(\left(1 + x \cdot 2.5\right) - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -6.8e-9)
   (+ wj (* (- wj (/ x (exp wj))) (/ -1.0 (+ wj 1.0))))
   (if (<= wj 8.6e-9)
     (+ x (* (pow wj 2.0) (- (+ 1.0 (* x 2.5)) wj)))
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -6.8e-9) {
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (wj + 1.0)));
	} else if (wj <= 8.6e-9) {
		tmp = x + (pow(wj, 2.0) * ((1.0 + (x * 2.5)) - wj));
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-6.8d-9)) then
        tmp = wj + ((wj - (x / exp(wj))) * ((-1.0d0) / (wj + 1.0d0)))
    else if (wj <= 8.6d-9) then
        tmp = x + ((wj ** 2.0d0) * ((1.0d0 + (x * 2.5d0)) - wj))
    else
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -6.8e-9) {
		tmp = wj + ((wj - (x / Math.exp(wj))) * (-1.0 / (wj + 1.0)));
	} else if (wj <= 8.6e-9) {
		tmp = x + (Math.pow(wj, 2.0) * ((1.0 + (x * 2.5)) - wj));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -6.8e-9:
		tmp = wj + ((wj - (x / math.exp(wj))) * (-1.0 / (wj + 1.0)))
	elif wj <= 8.6e-9:
		tmp = x + (math.pow(wj, 2.0) * ((1.0 + (x * 2.5)) - wj))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -6.8e-9)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) * Float64(-1.0 / Float64(wj + 1.0))));
	elseif (wj <= 8.6e-9)
		tmp = Float64(x + Float64((wj ^ 2.0) * Float64(Float64(1.0 + Float64(x * 2.5)) - wj)));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -6.8e-9)
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (wj + 1.0)));
	elseif (wj <= 8.6e-9)
		tmp = x + ((wj ^ 2.0) * ((1.0 + (x * 2.5)) - wj));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -6.8e-9], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 8.6e-9], N[(x + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -6.7999999999999997e-9

   1. Initial program 86.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 95.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 95.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 87.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 87.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 94.7%

         \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \frac{x}{e^{wj}}}}} \]

      2. associate-/r/ 95.4%

         \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)} \]

   6. Applied egg-rr 95.4%

      \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)} \]

   if -6.7999999999999997e-9 < wj < 8.59999999999999925e-9

   1. Initial program 76.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 76.2%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 76.2%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 76.2%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 76.2%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 76.2%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \color{blue}{{wj}^{3}} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \]

   7. Taylor expanded in wj around inf 99.7%

      \[\leadsto x + \color{blue}{\left(-1 \cdot {wj}^{3} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.7%

         \[\leadsto x + \left(\color{blue}{\left(-{wj}^{3}\right)} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      2. cube-mult 99.7%

         \[\leadsto x + \left(\left(-\color{blue}{wj \cdot \left(wj \cdot wj\right)}\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      3. unpow2 99.7%

         \[\leadsto x + \left(\left(-wj \cdot \color{blue}{{wj}^{2}}\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      4. distribute-lft-neg-out 99.7%

         \[\leadsto x + \left(\color{blue}{\left(-wj\right) \cdot {wj}^{2}} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      5. *-commutative 99.7%

         \[\leadsto x + \left(\left(-wj\right) \cdot {wj}^{2} + \color{blue}{\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}}\right) \]

      6. distribute-rgt-out 99.7%

         \[\leadsto x + \left(\left(-wj\right) \cdot {wj}^{2} + \left(1 - \color{blue}{x \cdot \left(-4 + 1.5\right)}\right) \cdot {wj}^{2}\right) \]

      7. metadata-eval 99.7%

         \[\leadsto x + \left(\left(-wj\right) \cdot {wj}^{2} + \left(1 - x \cdot \color{blue}{-2.5}\right) \cdot {wj}^{2}\right) \]

      8. distribute-rgt-out 99.7%

         \[\leadsto x + \color{blue}{{wj}^{2} \cdot \left(\left(-wj\right) + \left(1 - x \cdot -2.5\right)\right)} \]

      9. sub-neg 99.7%

         \[\leadsto x + {wj}^{2} \cdot \left(\left(-wj\right) + \color{blue}{\left(1 + \left(-x \cdot -2.5\right)\right)}\right) \]

      10. distribute-rgt-neg-in 99.7%

          \[\leadsto x + {wj}^{2} \cdot \left(\left(-wj\right) + \left(1 + \color{blue}{x \cdot \left(--2.5\right)}\right)\right) \]

      11. metadata-eval 99.7%

          \[\leadsto x + {wj}^{2} \cdot \left(\left(-wj\right) + \left(1 + x \cdot \color{blue}{2.5}\right)\right) \]

   9. Simplified 99.7%

      \[\leadsto x + \color{blue}{{wj}^{2} \cdot \left(\left(-wj\right) + \left(1 + x \cdot 2.5\right)\right)} \]

   if 8.59999999999999925e-9 < wj

   1. Initial program 46.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 46.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 46.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 46.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 46.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 96.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 96.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 96.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 96.6%

         \[\leadsto wj - \frac{wj - \color{blue}{\frac{1}{\frac{e^{wj}}{x}}}}{wj + 1} \]

      2. associate-/r/ 96.4%

         \[\leadsto wj - \frac{wj - \color{blue}{\frac{1}{e^{wj}} \cdot x}}{wj + 1} \]

      3. rec-exp 96.6%

         \[\leadsto wj - \frac{wj - \color{blue}{e^{-wj}} \cdot x}{wj + 1} \]

   6. Applied egg-rr 96.6%

      \[\leadsto wj - \frac{wj - \color{blue}{e^{-wj} \cdot x}}{wj + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.8 \cdot 10^{-9}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \mathbf{elif}\;wj \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(\left(1 + x \cdot 2.5\right) - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.8 \cdot 10^{-9} \lor \neg \left(wj \leq 6 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -5.8e-9) (not (<= wj 6e-9)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (+ (* -2.0 (* wj x)) (pow wj 2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -5.8e-9) || !(wj <= 6e-9)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + pow(wj, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-5.8d-9)) .or. (.not. (wj <= 6d-9))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + (wj ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -5.8e-9) || !(wj <= 6e-9)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + Math.pow(wj, 2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -5.8e-9) or not (wj <= 6e-9):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + math.pow(wj, 2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -5.8e-9) || !(wj <= 6e-9))
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + (wj ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -5.8e-9) || ~((wj <= 6e-9)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + (wj ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -5.8e-9], N[Not[LessEqual[wj, 6e-9]], $MachinePrecision]], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -5.79999999999999982e-9 or 5.99999999999999996e-9 < wj

   1. Initial program 68.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 73.0%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 73.1%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 68.5%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 68.5%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.8%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.8%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.8%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   if -5.79999999999999982e-9 < wj < 5.99999999999999996e-9

   1. Initial program 76.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 76.2%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 76.2%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 76.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 76.2%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 76.2%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 76.2%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.3%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]

   6. Taylor expanded in x around 0 99.3%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \color{blue}{{wj}^{2}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.8 \cdot 10^{-9} \lor \neg \left(wj \leq 6 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{\frac{1}{wj}}{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.2e-6)
   (+ x (+ (* -2.0 (* wj x)) (pow wj 2.0)))
   (/ 1.0 (+ (/ 1.0 wj) (/ (/ 1.0 wj) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.2e-6) {
		tmp = x + ((-2.0 * (wj * x)) + pow(wj, 2.0));
	} else {
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.2d-6) then
        tmp = x + (((-2.0d0) * (wj * x)) + (wj ** 2.0d0))
    else
        tmp = 1.0d0 / ((1.0d0 / wj) + ((1.0d0 / wj) / wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.2e-6) {
		tmp = x + ((-2.0 * (wj * x)) + Math.pow(wj, 2.0));
	} else {
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.2e-6:
		tmp = x + ((-2.0 * (wj * x)) + math.pow(wj, 2.0))
	else:
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.2e-6)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + (wj ^ 2.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / wj) + Float64(Float64(1.0 / wj) / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.2e-6)
		tmp = x + ((-2.0 * (wj * x)) + (wj ^ 2.0));
	else
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.2e-6], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / wj), $MachinePrecision] + N[(N[(1.0 / wj), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.2000000000000001e-6

   1. Initial program 77.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 77.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 77.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 77.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 77.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 77.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 77.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 97.0%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]

   6. Taylor expanded in x around 0 96.5%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \color{blue}{{wj}^{2}}\right) \]

   if 2.2000000000000001e-6 < wj

   1. Initial program 23.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 23.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 23.9%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 95.5%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 95.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 95.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 95.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in wj around inf 99.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{wj} + \frac{1}{{wj}^{2}}}} \]
   8. Step-by-step derivation

      1. inv-pow 99.3%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{\left({wj}^{2}\right)}^{-1}}} \]

      2. unpow2 99.3%

         \[\leadsto \frac{1}{\frac{1}{wj} + {\color{blue}{\left(wj \cdot wj\right)}}^{-1}} \]

      3. pow-prod-down 99.0%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{wj}^{-1} \cdot {wj}^{-1}}} \]

      4. inv-pow 99.0%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj}} \cdot {wj}^{-1}} \]

      5. inv-pow 99.0%

         \[\leadsto \frac{1}{\frac{1}{wj} + \frac{1}{wj} \cdot \color{blue}{\frac{1}{wj}}} \]

   9. Applied egg-rr 99.0%

      \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj} \cdot \frac{1}{wj}}} \]
   10. Step-by-step derivation

       1. un-div-inv 99.3%

          \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{\frac{1}{wj}}{wj}}} \]

   11. Applied egg-rr 99.3%

       \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{\frac{1}{wj}}{wj}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{\frac{1}{wj}}{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.1 \cdot 10^{-10}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right) - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;x + {wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{\frac{1}{wj}}{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.1e-10)
   (+ wj (/ (- (* x (- 1.0 wj)) wj) (+ wj 1.0)))
   (if (<= wj 1.75e-6)
     (+ x (pow wj 2.0))
     (/ 1.0 (+ (/ 1.0 wj) (/ (/ 1.0 wj) wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.1e-10) {
		tmp = wj + (((x * (1.0 - wj)) - wj) / (wj + 1.0));
	} else if (wj <= 1.75e-6) {
		tmp = x + pow(wj, 2.0);
	} else {
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3.1d-10)) then
        tmp = wj + (((x * (1.0d0 - wj)) - wj) / (wj + 1.0d0))
    else if (wj <= 1.75d-6) then
        tmp = x + (wj ** 2.0d0)
    else
        tmp = 1.0d0 / ((1.0d0 / wj) + ((1.0d0 / wj) / wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.1e-10) {
		tmp = wj + (((x * (1.0 - wj)) - wj) / (wj + 1.0));
	} else if (wj <= 1.75e-6) {
		tmp = x + Math.pow(wj, 2.0);
	} else {
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.1e-10:
		tmp = wj + (((x * (1.0 - wj)) - wj) / (wj + 1.0))
	elif wj <= 1.75e-6:
		tmp = x + math.pow(wj, 2.0)
	else:
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.1e-10)
		tmp = Float64(wj + Float64(Float64(Float64(x * Float64(1.0 - wj)) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 1.75e-6)
		tmp = Float64(x + (wj ^ 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / wj) + Float64(Float64(1.0 / wj) / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.1e-10)
		tmp = wj + (((x * (1.0 - wj)) - wj) / (wj + 1.0));
	elseif (wj <= 1.75e-6)
		tmp = x + (wj ^ 2.0);
	else
		tmp = 1.0 / ((1.0 / wj) + ((1.0 / wj) / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.1e-10], N[(wj + N[(N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.75e-6], N[(x + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / wj), $MachinePrecision] + N[(N[(1.0 / wj), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -3.10000000000000015e-10

   1. Initial program 85.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 92.8%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 93.1%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 86.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 86.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 93.1%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 93.1%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 72.5%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + -1 \cdot \left(wj \cdot x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 72.5%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-1 \cdot wj\right) \cdot x}\right)}{wj + 1} \]

      2. neg-mul-1 72.5%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-wj\right)} \cdot x\right)}{wj + 1} \]

      3. distribute-rgt1-in 72.5%

         \[\leadsto wj - \frac{wj - \color{blue}{\left(\left(-wj\right) + 1\right) \cdot x}}{wj + 1} \]

      4. +-commutative 72.5%

         \[\leadsto wj - \frac{wj - \color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x}{wj + 1} \]

      5. sub-neg 72.5%

         \[\leadsto wj - \frac{wj - \color{blue}{\left(1 - wj\right)} \cdot x}{wj + 1} \]

   7. Simplified 72.5%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(1 - wj\right) \cdot x}}{wj + 1} \]

   if -3.10000000000000015e-10 < wj < 1.74999999999999997e-6

   1. Initial program 76.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 76.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 76.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 76.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 76.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 76.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 76.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 76.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.4%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]

   6. Taylor expanded in x around 0 99.3%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \color{blue}{{wj}^{2}}\right) \]

   7. Taylor expanded in wj around inf 98.7%

      \[\leadsto x + \color{blue}{{wj}^{2}} \]

   if 1.74999999999999997e-6 < wj

   1. Initial program 23.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 23.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 23.9%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 95.5%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 95.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 95.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 95.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in wj around inf 99.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{wj} + \frac{1}{{wj}^{2}}}} \]
   8. Step-by-step derivation

      1. inv-pow 99.3%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{\left({wj}^{2}\right)}^{-1}}} \]

      2. unpow2 99.3%

         \[\leadsto \frac{1}{\frac{1}{wj} + {\color{blue}{\left(wj \cdot wj\right)}}^{-1}} \]

      3. pow-prod-down 99.0%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{wj}^{-1} \cdot {wj}^{-1}}} \]

      4. inv-pow 99.0%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj}} \cdot {wj}^{-1}} \]

      5. inv-pow 99.0%

         \[\leadsto \frac{1}{\frac{1}{wj} + \frac{1}{wj} \cdot \color{blue}{\frac{1}{wj}}} \]

   9. Applied egg-rr 99.0%

      \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj} \cdot \frac{1}{wj}}} \]
   10. Step-by-step derivation

       1. un-div-inv 99.3%

          \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{\frac{1}{wj}}{wj}}} \]

   11. Applied egg-rr 99.3%

       \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{\frac{1}{wj}}{wj}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.1 \cdot 10^{-10}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right) - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;x + {wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{\frac{1}{wj}}{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.7% accurate, 19.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-153}:\\ \;\;\;\;wj + \frac{1 - wj}{\frac{wj + 1}{x}}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-230}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{-1}{wj \cdot \left(-wj\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -6.5e-153)
   (+ wj (/ (- 1.0 wj) (/ (+ wj 1.0) x)))
   (if (<= x 6.6e-230)
     (/ 1.0 (+ (/ 1.0 wj) (/ -1.0 (* wj (- wj)))))
     (/ x (/ (+ wj 1.0) (- 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -6.5e-153) {
		tmp = wj + ((1.0 - wj) / ((wj + 1.0) / x));
	} else if (x <= 6.6e-230) {
		tmp = 1.0 / ((1.0 / wj) + (-1.0 / (wj * -wj)));
	} else {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-6.5d-153)) then
        tmp = wj + ((1.0d0 - wj) / ((wj + 1.0d0) / x))
    else if (x <= 6.6d-230) then
        tmp = 1.0d0 / ((1.0d0 / wj) + ((-1.0d0) / (wj * -wj)))
    else
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -6.5e-153) {
		tmp = wj + ((1.0 - wj) / ((wj + 1.0) / x));
	} else if (x <= 6.6e-230) {
		tmp = 1.0 / ((1.0 / wj) + (-1.0 / (wj * -wj)));
	} else {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -6.5e-153:
		tmp = wj + ((1.0 - wj) / ((wj + 1.0) / x))
	elif x <= 6.6e-230:
		tmp = 1.0 / ((1.0 / wj) + (-1.0 / (wj * -wj)))
	else:
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -6.5e-153)
		tmp = Float64(wj + Float64(Float64(1.0 - wj) / Float64(Float64(wj + 1.0) / x)));
	elseif (x <= 6.6e-230)
		tmp = Float64(1.0 / Float64(Float64(1.0 / wj) + Float64(-1.0 / Float64(wj * Float64(-wj)))));
	else
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -6.5e-153)
		tmp = wj + ((1.0 - wj) / ((wj + 1.0) / x));
	elseif (x <= 6.6e-230)
		tmp = 1.0 / ((1.0 / wj) + (-1.0 / (wj * -wj)));
	else
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -6.5e-153], N[(wj + N[(N[(1.0 - wj), $MachinePrecision] / N[(N[(wj + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-230], N[(1.0 / N[(N[(1.0 / wj), $MachinePrecision] + N[(-1.0 / N[(wj * (-wj)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.50000000000000032e-153

   1. Initial program 94.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 93.9%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 94.0%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 94.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 94.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 98.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 98.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 94.6%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + -1 \cdot \left(wj \cdot x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 94.6%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-1 \cdot wj\right) \cdot x}\right)}{wj + 1} \]

      2. neg-mul-1 94.6%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-wj\right)} \cdot x\right)}{wj + 1} \]

   7. Simplified 94.6%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + \left(-wj\right) \cdot x\right)}}{wj + 1} \]

   8. Taylor expanded in x around inf 92.0%

      \[\leadsto wj - \color{blue}{-1 \cdot \frac{x \cdot \left(1 + -1 \cdot wj\right)}{1 + wj}} \]
   9. Step-by-step derivation

      1. mul-1-neg 92.0%

         \[\leadsto wj - \color{blue}{\left(-\frac{x \cdot \left(1 + -1 \cdot wj\right)}{1 + wj}\right)} \]

      2. neg-mul-1 92.0%

         \[\leadsto wj - \left(-\frac{x \cdot \left(1 + \color{blue}{\left(-wj\right)}\right)}{1 + wj}\right) \]

      3. sub-neg 92.0%

         \[\leadsto wj - \left(-\frac{x \cdot \color{blue}{\left(1 - wj\right)}}{1 + wj}\right) \]

      4. *-commutative 92.0%

         \[\leadsto wj - \left(-\frac{\color{blue}{\left(1 - wj\right) \cdot x}}{1 + wj}\right) \]

      5. sub-neg 92.0%

         \[\leadsto wj - \left(-\frac{\color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x}{1 + wj}\right) \]

      6. +-commutative 92.0%

         \[\leadsto wj - \left(-\frac{\color{blue}{\left(\left(-wj\right) + 1\right)} \cdot x}{1 + wj}\right) \]

      7. neg-mul-1 92.0%

         \[\leadsto wj - \left(-\frac{\left(\color{blue}{-1 \cdot wj} + 1\right) \cdot x}{1 + wj}\right) \]

      8. fma-udef 92.0%

         \[\leadsto wj - \left(-\frac{\color{blue}{\mathsf{fma}\left(-1, wj, 1\right)} \cdot x}{1 + wj}\right) \]

      9. +-commutative 92.0%

         \[\leadsto wj - \left(-\frac{\mathsf{fma}\left(-1, wj, 1\right) \cdot x}{\color{blue}{wj + 1}}\right) \]

      10. associate-/l* 91.6%

          \[\leadsto wj - \left(-\color{blue}{\frac{\mathsf{fma}\left(-1, wj, 1\right)}{\frac{wj + 1}{x}}}\right) \]

      11. fma-udef 91.6%

          \[\leadsto wj - \left(-\frac{\color{blue}{-1 \cdot wj + 1}}{\frac{wj + 1}{x}}\right) \]

      12. neg-mul-1 91.6%

          \[\leadsto wj - \left(-\frac{\color{blue}{\left(-wj\right)} + 1}{\frac{wj + 1}{x}}\right) \]

      13. +-commutative 91.6%

          \[\leadsto wj - \left(-\frac{\color{blue}{1 + \left(-wj\right)}}{\frac{wj + 1}{x}}\right) \]

      14. sub-neg 91.6%

          \[\leadsto wj - \left(-\frac{\color{blue}{1 - wj}}{\frac{wj + 1}{x}}\right) \]

   10. Simplified 91.6%

       \[\leadsto wj - \color{blue}{\left(-\frac{1 - wj}{\frac{wj + 1}{x}}\right)} \]

   if -6.50000000000000032e-153 < x < 6.59999999999999987e-230

   1. Initial program 31.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 33.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 33.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 35.6%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 35.6%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 35.6%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 21.3%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 21.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in wj around inf 63.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{wj} + \frac{1}{{wj}^{2}}}} \]
   8. Step-by-step derivation

      1. inv-pow 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{\left({wj}^{2}\right)}^{-1}}} \]

      2. unpow2 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + {\color{blue}{\left(wj \cdot wj\right)}}^{-1}} \]

      3. pow-prod-down 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{wj}^{-1} \cdot {wj}^{-1}}} \]

      4. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj}} \cdot {wj}^{-1}} \]

      5. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \frac{1}{wj} \cdot \color{blue}{\frac{1}{wj}}} \]

   9. Applied egg-rr 63.5%

      \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj} \cdot \frac{1}{wj}}} \]
   10. Step-by-step derivation

       1. frac-2neg 63.5%

          \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{-1}{-wj}} \cdot \frac{1}{wj}} \]

       2. metadata-eval 63.5%

          \[\leadsto \frac{1}{\frac{1}{wj} + \frac{\color{blue}{-1}}{-wj} \cdot \frac{1}{wj}} \]

       3. frac-times 63.9%

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

       4. metadata-eval 63.9%

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

   11. Applied egg-rr 63.9%

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

   if 6.59999999999999987e-230 < x

   1. Initial program 81.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 81.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 81.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 81.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 81.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 81.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 81.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 80.8%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + -1 \cdot \left(wj \cdot x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 80.8%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-1 \cdot wj\right) \cdot x}\right)}{wj + 1} \]

      2. neg-mul-1 80.8%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-wj\right)} \cdot x\right)}{wj + 1} \]

   7. Simplified 80.8%

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

      1. clear-num 80.6%

         \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \left(x + \left(-wj\right) \cdot x\right)}}} \]

      2. inv-pow 80.6%

         \[\leadsto wj - \color{blue}{{\left(\frac{wj + 1}{wj - \left(x + \left(-wj\right) \cdot x\right)}\right)}^{-1}} \]

      3. distribute-rgt1-in 80.6%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \color{blue}{\left(\left(-wj\right) + 1\right) \cdot x}}\right)}^{-1} \]

      4. neg-mul-1 80.6%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \left(\color{blue}{-1 \cdot wj} + 1\right) \cdot x}\right)}^{-1} \]

      5. fma-def 80.6%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \color{blue}{\mathsf{fma}\left(-1, wj, 1\right)} \cdot x}\right)}^{-1} \]

   9. Applied egg-rr 80.6%

      \[\leadsto wj - \color{blue}{{\left(\frac{wj + 1}{wj - \mathsf{fma}\left(-1, wj, 1\right) \cdot x}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 80.6%

          \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \mathsf{fma}\left(-1, wj, 1\right) \cdot x}}} \]

       2. fma-udef 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(-1 \cdot wj + 1\right)} \cdot x}} \]

       3. neg-mul-1 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \left(\color{blue}{\left(-wj\right)} + 1\right) \cdot x}} \]

       4. +-commutative 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x}} \]

       5. sub-neg 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(1 - wj\right)} \cdot x}} \]

       6. *-commutative 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{x \cdot \left(1 - wj\right)}}} \]

   11. Simplified 80.6%

       \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - x \cdot \left(1 - wj\right)}}} \]

   12. Taylor expanded in x around -inf 90.7%

       \[\leadsto \color{blue}{\frac{x \cdot \left(1 - wj\right)}{1 + wj}} \]
   13. Step-by-step derivation

       1. associate-/l* 90.7%

          \[\leadsto \color{blue}{\frac{x}{\frac{1 + wj}{1 - wj}}} \]

       2. +-commutative 90.7%

          \[\leadsto \frac{x}{\frac{\color{blue}{wj + 1}}{1 - wj}} \]

   14. Simplified 90.7%

       \[\leadsto \color{blue}{\frac{x}{\frac{wj + 1}{1 - wj}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-153}:\\ \;\;\;\;wj + \frac{1 - wj}{\frac{wj + 1}{x}}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-230}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{-1}{wj \cdot \left(-wj\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.6% accurate, 19.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-153}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right) - wj}{wj + 1}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-230}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{-1}{wj \cdot \left(-wj\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -3.9e-153)
   (+ wj (/ (- (* x (- 1.0 wj)) wj) (+ wj 1.0)))
   (if (<= x 5.6e-230)
     (/ 1.0 (+ (/ 1.0 wj) (/ -1.0 (* wj (- wj)))))
     (/ x (/ (+ wj 1.0) (- 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -3.9e-153) {
		tmp = wj + (((x * (1.0 - wj)) - wj) / (wj + 1.0));
	} else if (x <= 5.6e-230) {
		tmp = 1.0 / ((1.0 / wj) + (-1.0 / (wj * -wj)));
	} else {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.9d-153)) then
        tmp = wj + (((x * (1.0d0 - wj)) - wj) / (wj + 1.0d0))
    else if (x <= 5.6d-230) then
        tmp = 1.0d0 / ((1.0d0 / wj) + ((-1.0d0) / (wj * -wj)))
    else
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -3.9e-153) {
		tmp = wj + (((x * (1.0 - wj)) - wj) / (wj + 1.0));
	} else if (x <= 5.6e-230) {
		tmp = 1.0 / ((1.0 / wj) + (-1.0 / (wj * -wj)));
	} else {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -3.9e-153:
		tmp = wj + (((x * (1.0 - wj)) - wj) / (wj + 1.0))
	elif x <= 5.6e-230:
		tmp = 1.0 / ((1.0 / wj) + (-1.0 / (wj * -wj)))
	else:
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -3.9e-153)
		tmp = Float64(wj + Float64(Float64(Float64(x * Float64(1.0 - wj)) - wj) / Float64(wj + 1.0)));
	elseif (x <= 5.6e-230)
		tmp = Float64(1.0 / Float64(Float64(1.0 / wj) + Float64(-1.0 / Float64(wj * Float64(-wj)))));
	else
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -3.9e-153)
		tmp = wj + (((x * (1.0 - wj)) - wj) / (wj + 1.0));
	elseif (x <= 5.6e-230)
		tmp = 1.0 / ((1.0 / wj) + (-1.0 / (wj * -wj)));
	else
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -3.9e-153], N[(wj + N[(N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-230], N[(1.0 / N[(N[(1.0 / wj), $MachinePrecision] + N[(-1.0 / N[(wj * (-wj)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9000000000000002e-153

   1. Initial program 94.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 93.9%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 94.0%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 94.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 94.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 98.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 98.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 94.6%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + -1 \cdot \left(wj \cdot x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 94.6%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-1 \cdot wj\right) \cdot x}\right)}{wj + 1} \]

      2. neg-mul-1 94.6%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-wj\right)} \cdot x\right)}{wj + 1} \]

      3. distribute-rgt1-in 94.6%

         \[\leadsto wj - \frac{wj - \color{blue}{\left(\left(-wj\right) + 1\right) \cdot x}}{wj + 1} \]

      4. +-commutative 94.6%

         \[\leadsto wj - \frac{wj - \color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x}{wj + 1} \]

      5. sub-neg 94.6%

         \[\leadsto wj - \frac{wj - \color{blue}{\left(1 - wj\right)} \cdot x}{wj + 1} \]

   7. Simplified 94.6%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(1 - wj\right) \cdot x}}{wj + 1} \]

   if -3.9000000000000002e-153 < x < 5.6000000000000002e-230

   1. Initial program 31.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 33.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 33.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 35.6%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 35.6%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 35.6%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 21.3%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 21.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in wj around inf 63.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{wj} + \frac{1}{{wj}^{2}}}} \]
   8. Step-by-step derivation

      1. inv-pow 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{\left({wj}^{2}\right)}^{-1}}} \]

      2. unpow2 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + {\color{blue}{\left(wj \cdot wj\right)}}^{-1}} \]

      3. pow-prod-down 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{wj}^{-1} \cdot {wj}^{-1}}} \]

      4. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj}} \cdot {wj}^{-1}} \]

      5. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \frac{1}{wj} \cdot \color{blue}{\frac{1}{wj}}} \]

   9. Applied egg-rr 63.5%

      \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj} \cdot \frac{1}{wj}}} \]
   10. Step-by-step derivation

       1. frac-2neg 63.5%

          \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{-1}{-wj}} \cdot \frac{1}{wj}} \]

       2. metadata-eval 63.5%

          \[\leadsto \frac{1}{\frac{1}{wj} + \frac{\color{blue}{-1}}{-wj} \cdot \frac{1}{wj}} \]

       3. frac-times 63.9%

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

       4. metadata-eval 63.9%

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

   11. Applied egg-rr 63.9%

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

   if 5.6000000000000002e-230 < x

   1. Initial program 81.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 81.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 81.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 81.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 81.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 81.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 81.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 80.8%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + -1 \cdot \left(wj \cdot x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 80.8%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-1 \cdot wj\right) \cdot x}\right)}{wj + 1} \]

      2. neg-mul-1 80.8%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-wj\right)} \cdot x\right)}{wj + 1} \]

   7. Simplified 80.8%

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

      1. clear-num 80.6%

         \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \left(x + \left(-wj\right) \cdot x\right)}}} \]

      2. inv-pow 80.6%

         \[\leadsto wj - \color{blue}{{\left(\frac{wj + 1}{wj - \left(x + \left(-wj\right) \cdot x\right)}\right)}^{-1}} \]

      3. distribute-rgt1-in 80.6%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \color{blue}{\left(\left(-wj\right) + 1\right) \cdot x}}\right)}^{-1} \]

      4. neg-mul-1 80.6%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \left(\color{blue}{-1 \cdot wj} + 1\right) \cdot x}\right)}^{-1} \]

      5. fma-def 80.6%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \color{blue}{\mathsf{fma}\left(-1, wj, 1\right)} \cdot x}\right)}^{-1} \]

   9. Applied egg-rr 80.6%

      \[\leadsto wj - \color{blue}{{\left(\frac{wj + 1}{wj - \mathsf{fma}\left(-1, wj, 1\right) \cdot x}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 80.6%

          \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \mathsf{fma}\left(-1, wj, 1\right) \cdot x}}} \]

       2. fma-udef 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(-1 \cdot wj + 1\right)} \cdot x}} \]

       3. neg-mul-1 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \left(\color{blue}{\left(-wj\right)} + 1\right) \cdot x}} \]

       4. +-commutative 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x}} \]

       5. sub-neg 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(1 - wj\right)} \cdot x}} \]

       6. *-commutative 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{x \cdot \left(1 - wj\right)}}} \]

   11. Simplified 80.6%

       \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - x \cdot \left(1 - wj\right)}}} \]

   12. Taylor expanded in x around -inf 90.7%

       \[\leadsto \color{blue}{\frac{x \cdot \left(1 - wj\right)}{1 + wj}} \]
   13. Step-by-step derivation

       1. associate-/l* 90.7%

          \[\leadsto \color{blue}{\frac{x}{\frac{1 + wj}{1 - wj}}} \]

       2. +-commutative 90.7%

          \[\leadsto \frac{x}{\frac{\color{blue}{wj + 1}}{1 - wj}} \]

   14. Simplified 90.7%

       \[\leadsto \color{blue}{\frac{x}{\frac{wj + 1}{1 - wj}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-153}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right) - wj}{wj + 1}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-230}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{-1}{wj \cdot \left(-wj\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.0% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-153} \lor \neg \left(x \leq 6 \cdot 10^{-228}\right):\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \frac{1}{1 + \frac{1}{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -3.9e-153) (not (<= x 6e-228)))
   (/ x (/ (+ wj 1.0) (- 1.0 wj)))
   (* wj (/ 1.0 (+ 1.0 (/ 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -3.9e-153) || !(x <= 6e-228)) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} else {
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.9d-153)) .or. (.not. (x <= 6d-228))) then
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    else
        tmp = wj * (1.0d0 / (1.0d0 + (1.0d0 / wj)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -3.9e-153) || !(x <= 6e-228)) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} else {
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -3.9e-153) or not (x <= 6e-228):
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	else:
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -3.9e-153) || !(x <= 6e-228))
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	else
		tmp = Float64(wj * Float64(1.0 / Float64(1.0 + Float64(1.0 / wj))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -3.9e-153) || ~((x <= 6e-228)))
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	else
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -3.9e-153], N[Not[LessEqual[x, 6e-228]], $MachinePrecision]], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(1.0 / N[(1.0 + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9000000000000002e-153 or 5.9999999999999999e-228 < x

   1. Initial program 87.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 87.2%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 87.2%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 87.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 87.2%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 89.2%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 89.2%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 87.2%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + -1 \cdot \left(wj \cdot x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 87.2%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-1 \cdot wj\right) \cdot x}\right)}{wj + 1} \]

      2. neg-mul-1 87.2%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-wj\right)} \cdot x\right)}{wj + 1} \]

   7. Simplified 87.2%

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

      1. clear-num 86.8%

         \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \left(x + \left(-wj\right) \cdot x\right)}}} \]

      2. inv-pow 86.8%

         \[\leadsto wj - \color{blue}{{\left(\frac{wj + 1}{wj - \left(x + \left(-wj\right) \cdot x\right)}\right)}^{-1}} \]

      3. distribute-rgt1-in 86.8%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \color{blue}{\left(\left(-wj\right) + 1\right) \cdot x}}\right)}^{-1} \]

      4. neg-mul-1 86.8%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \left(\color{blue}{-1 \cdot wj} + 1\right) \cdot x}\right)}^{-1} \]

      5. fma-def 86.8%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \color{blue}{\mathsf{fma}\left(-1, wj, 1\right)} \cdot x}\right)}^{-1} \]

   9. Applied egg-rr 86.8%

      \[\leadsto wj - \color{blue}{{\left(\frac{wj + 1}{wj - \mathsf{fma}\left(-1, wj, 1\right) \cdot x}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 86.8%

          \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \mathsf{fma}\left(-1, wj, 1\right) \cdot x}}} \]

       2. fma-udef 86.8%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(-1 \cdot wj + 1\right)} \cdot x}} \]

       3. neg-mul-1 86.8%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \left(\color{blue}{\left(-wj\right)} + 1\right) \cdot x}} \]

       4. +-commutative 86.8%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x}} \]

       5. sub-neg 86.8%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(1 - wj\right)} \cdot x}} \]

       6. *-commutative 86.8%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{x \cdot \left(1 - wj\right)}}} \]

   11. Simplified 86.8%

       \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - x \cdot \left(1 - wj\right)}}} \]

   12. Taylor expanded in x around -inf 90.9%

       \[\leadsto \color{blue}{\frac{x \cdot \left(1 - wj\right)}{1 + wj}} \]
   13. Step-by-step derivation

       1. associate-/l* 90.9%

          \[\leadsto \color{blue}{\frac{x}{\frac{1 + wj}{1 - wj}}} \]

       2. +-commutative 90.9%

          \[\leadsto \frac{x}{\frac{\color{blue}{wj + 1}}{1 - wj}} \]

   14. Simplified 90.9%

       \[\leadsto \color{blue}{\frac{x}{\frac{wj + 1}{1 - wj}}} \]

   if -3.9000000000000002e-153 < x < 5.9999999999999999e-228

   1. Initial program 31.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 33.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 33.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 35.6%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 35.6%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 35.6%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 21.3%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 21.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in wj around inf 63.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{wj} + \frac{1}{{wj}^{2}}}} \]
   8. Step-by-step derivation

      1. inv-pow 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{\left({wj}^{2}\right)}^{-1}}} \]

      2. unpow2 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + {\color{blue}{\left(wj \cdot wj\right)}}^{-1}} \]

      3. pow-prod-down 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{wj}^{-1} \cdot {wj}^{-1}}} \]

      4. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj}} \cdot {wj}^{-1}} \]

      5. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \frac{1}{wj} \cdot \color{blue}{\frac{1}{wj}}} \]

   9. Applied egg-rr 63.5%

      \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj} \cdot \frac{1}{wj}}} \]
   10. Step-by-step derivation

       1. inv-pow 63.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{wj} + \frac{1}{wj} \cdot \frac{1}{wj}\right)}^{-1}} \]

       2. distribute-rgt1-in 63.5%

          \[\leadsto {\color{blue}{\left(\left(\frac{1}{wj} + 1\right) \cdot \frac{1}{wj}\right)}}^{-1} \]

       3. unpow-prod-down 63.7%

          \[\leadsto \color{blue}{{\left(\frac{1}{wj} + 1\right)}^{-1} \cdot {\left(\frac{1}{wj}\right)}^{-1}} \]

       4. +-commutative 63.7%

          \[\leadsto {\color{blue}{\left(1 + \frac{1}{wj}\right)}}^{-1} \cdot {\left(\frac{1}{wj}\right)}^{-1} \]

   11. Applied egg-rr 63.7%

       \[\leadsto \color{blue}{{\left(1 + \frac{1}{wj}\right)}^{-1} \cdot {\left(\frac{1}{wj}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 63.7%

          \[\leadsto {\left(1 + \frac{1}{wj}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{1}{wj}}} \]

       2. remove-double-div 63.8%

          \[\leadsto {\left(1 + \frac{1}{wj}\right)}^{-1} \cdot \color{blue}{wj} \]

       3. *-commutative 63.8%

          \[\leadsto \color{blue}{wj \cdot {\left(1 + \frac{1}{wj}\right)}^{-1}} \]

       4. unpow-1 63.8%

          \[\leadsto wj \cdot \color{blue}{\frac{1}{1 + \frac{1}{wj}}} \]

   13. Simplified 63.8%

       \[\leadsto \color{blue}{wj \cdot \frac{1}{1 + \frac{1}{wj}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-153} \lor \neg \left(x \leq 6 \cdot 10^{-228}\right):\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \frac{1}{1 + \frac{1}{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.0% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-153}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-227}:\\ \;\;\;\;wj \cdot \frac{1}{1 + \frac{1}{wj}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -3.9e-153)
   (+ x (* -2.0 (* wj x)))
   (if (<= x 1.55e-227)
     (* wj (/ 1.0 (+ 1.0 (/ 1.0 wj))))
     (/ x (+ 1.0 (* wj 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -3.9e-153) {
		tmp = x + (-2.0 * (wj * x));
	} else if (x <= 1.55e-227) {
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)));
	} else {
		tmp = x / (1.0 + (wj * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.9d-153)) then
        tmp = x + ((-2.0d0) * (wj * x))
    else if (x <= 1.55d-227) then
        tmp = wj * (1.0d0 / (1.0d0 + (1.0d0 / wj)))
    else
        tmp = x / (1.0d0 + (wj * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -3.9e-153) {
		tmp = x + (-2.0 * (wj * x));
	} else if (x <= 1.55e-227) {
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)));
	} else {
		tmp = x / (1.0 + (wj * 2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -3.9e-153:
		tmp = x + (-2.0 * (wj * x))
	elif x <= 1.55e-227:
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)))
	else:
		tmp = x / (1.0 + (wj * 2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -3.9e-153)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	elseif (x <= 1.55e-227)
		tmp = Float64(wj * Float64(1.0 / Float64(1.0 + Float64(1.0 / wj))));
	else
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -3.9e-153)
		tmp = x + (-2.0 * (wj * x));
	elseif (x <= 1.55e-227)
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)));
	else
		tmp = x / (1.0 + (wj * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -3.9e-153], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-227], N[(wj * N[(1.0 / N[(1.0 + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9000000000000002e-153

   1. Initial program 94.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 93.9%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 94.0%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 94.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 94.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 98.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 98.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 91.0%

      \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 91.0%

         \[\leadsto x + -2 \cdot \color{blue}{\left(x \cdot wj\right)} \]

   7. Simplified 91.0%

      \[\leadsto \color{blue}{x + -2 \cdot \left(x \cdot wj\right)} \]

   if -3.9000000000000002e-153 < x < 1.5499999999999999e-227

   1. Initial program 31.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 33.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 33.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 35.6%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 35.6%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 35.6%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 21.3%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 21.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in wj around inf 63.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{wj} + \frac{1}{{wj}^{2}}}} \]
   8. Step-by-step derivation

      1. inv-pow 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{\left({wj}^{2}\right)}^{-1}}} \]

      2. unpow2 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + {\color{blue}{\left(wj \cdot wj\right)}}^{-1}} \]

      3. pow-prod-down 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{wj}^{-1} \cdot {wj}^{-1}}} \]

      4. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj}} \cdot {wj}^{-1}} \]

      5. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \frac{1}{wj} \cdot \color{blue}{\frac{1}{wj}}} \]

   9. Applied egg-rr 63.5%

      \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj} \cdot \frac{1}{wj}}} \]
   10. Step-by-step derivation

       1. inv-pow 63.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{wj} + \frac{1}{wj} \cdot \frac{1}{wj}\right)}^{-1}} \]

       2. distribute-rgt1-in 63.5%

          \[\leadsto {\color{blue}{\left(\left(\frac{1}{wj} + 1\right) \cdot \frac{1}{wj}\right)}}^{-1} \]

       3. unpow-prod-down 63.7%

          \[\leadsto \color{blue}{{\left(\frac{1}{wj} + 1\right)}^{-1} \cdot {\left(\frac{1}{wj}\right)}^{-1}} \]

       4. +-commutative 63.7%

          \[\leadsto {\color{blue}{\left(1 + \frac{1}{wj}\right)}}^{-1} \cdot {\left(\frac{1}{wj}\right)}^{-1} \]

   11. Applied egg-rr 63.7%

       \[\leadsto \color{blue}{{\left(1 + \frac{1}{wj}\right)}^{-1} \cdot {\left(\frac{1}{wj}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 63.7%

          \[\leadsto {\left(1 + \frac{1}{wj}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{1}{wj}}} \]

       2. remove-double-div 63.8%

          \[\leadsto {\left(1 + \frac{1}{wj}\right)}^{-1} \cdot \color{blue}{wj} \]

       3. *-commutative 63.8%

          \[\leadsto \color{blue}{wj \cdot {\left(1 + \frac{1}{wj}\right)}^{-1}} \]

       4. unpow-1 63.8%

          \[\leadsto wj \cdot \color{blue}{\frac{1}{1 + \frac{1}{wj}}} \]

   13. Simplified 63.8%

       \[\leadsto \color{blue}{wj \cdot \frac{1}{1 + \frac{1}{wj}}} \]

   if 1.5499999999999999e-227 < x

   1. Initial program 81.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 81.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 81.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 81.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 81.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 81.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 81.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 55.5%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 55.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 55.4%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 55.4%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in x around inf 91.3%

      \[\leadsto \color{blue}{\frac{x}{e^{wj} \cdot \left(1 + wj\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 91.3%

         \[\leadsto \frac{x}{\color{blue}{\left(1 + wj\right) \cdot e^{wj}}} \]

      2. +-commutative 91.3%

         \[\leadsto \frac{x}{\color{blue}{\left(wj + 1\right)} \cdot e^{wj}} \]

   9. Simplified 91.3%

      \[\leadsto \color{blue}{\frac{x}{\left(wj + 1\right) \cdot e^{wj}}} \]

   10. Taylor expanded in wj around 0 90.6%

       \[\leadsto \frac{x}{\color{blue}{1 + 2 \cdot wj}} \]
   11. Step-by-step derivation

       1. *-commutative 90.6%

          \[\leadsto \frac{x}{1 + \color{blue}{wj \cdot 2}} \]

   12. Simplified 90.6%

       \[\leadsto \frac{x}{\color{blue}{1 + wj \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-153}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-227}:\\ \;\;\;\;wj \cdot \frac{1}{1 + \frac{1}{wj}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.7% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-152}:\\ \;\;\;\;wj + \frac{1 - wj}{\frac{wj + 1}{x}}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-230}:\\ \;\;\;\;wj \cdot \frac{1}{1 + \frac{1}{wj}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -3.7e-152)
   (+ wj (/ (- 1.0 wj) (/ (+ wj 1.0) x)))
   (if (<= x 6.2e-230)
     (* wj (/ 1.0 (+ 1.0 (/ 1.0 wj))))
     (/ x (/ (+ wj 1.0) (- 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -3.7e-152) {
		tmp = wj + ((1.0 - wj) / ((wj + 1.0) / x));
	} else if (x <= 6.2e-230) {
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)));
	} else {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.7d-152)) then
        tmp = wj + ((1.0d0 - wj) / ((wj + 1.0d0) / x))
    else if (x <= 6.2d-230) then
        tmp = wj * (1.0d0 / (1.0d0 + (1.0d0 / wj)))
    else
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -3.7e-152) {
		tmp = wj + ((1.0 - wj) / ((wj + 1.0) / x));
	} else if (x <= 6.2e-230) {
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)));
	} else {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -3.7e-152:
		tmp = wj + ((1.0 - wj) / ((wj + 1.0) / x))
	elif x <= 6.2e-230:
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)))
	else:
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -3.7e-152)
		tmp = Float64(wj + Float64(Float64(1.0 - wj) / Float64(Float64(wj + 1.0) / x)));
	elseif (x <= 6.2e-230)
		tmp = Float64(wj * Float64(1.0 / Float64(1.0 + Float64(1.0 / wj))));
	else
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -3.7e-152)
		tmp = wj + ((1.0 - wj) / ((wj + 1.0) / x));
	elseif (x <= 6.2e-230)
		tmp = wj * (1.0 / (1.0 + (1.0 / wj)));
	else
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -3.7e-152], N[(wj + N[(N[(1.0 - wj), $MachinePrecision] / N[(N[(wj + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e-230], N[(wj * N[(1.0 / N[(1.0 + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6999999999999998e-152

   1. Initial program 94.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 93.9%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 94.0%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 94.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 94.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 98.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 98.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 94.6%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + -1 \cdot \left(wj \cdot x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 94.6%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-1 \cdot wj\right) \cdot x}\right)}{wj + 1} \]

      2. neg-mul-1 94.6%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-wj\right)} \cdot x\right)}{wj + 1} \]

   7. Simplified 94.6%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + \left(-wj\right) \cdot x\right)}}{wj + 1} \]

   8. Taylor expanded in x around inf 92.0%

      \[\leadsto wj - \color{blue}{-1 \cdot \frac{x \cdot \left(1 + -1 \cdot wj\right)}{1 + wj}} \]
   9. Step-by-step derivation

      1. mul-1-neg 92.0%

         \[\leadsto wj - \color{blue}{\left(-\frac{x \cdot \left(1 + -1 \cdot wj\right)}{1 + wj}\right)} \]

      2. neg-mul-1 92.0%

         \[\leadsto wj - \left(-\frac{x \cdot \left(1 + \color{blue}{\left(-wj\right)}\right)}{1 + wj}\right) \]

      3. sub-neg 92.0%

         \[\leadsto wj - \left(-\frac{x \cdot \color{blue}{\left(1 - wj\right)}}{1 + wj}\right) \]

      4. *-commutative 92.0%

         \[\leadsto wj - \left(-\frac{\color{blue}{\left(1 - wj\right) \cdot x}}{1 + wj}\right) \]

      5. sub-neg 92.0%

         \[\leadsto wj - \left(-\frac{\color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x}{1 + wj}\right) \]

      6. +-commutative 92.0%

         \[\leadsto wj - \left(-\frac{\color{blue}{\left(\left(-wj\right) + 1\right)} \cdot x}{1 + wj}\right) \]

      7. neg-mul-1 92.0%

         \[\leadsto wj - \left(-\frac{\left(\color{blue}{-1 \cdot wj} + 1\right) \cdot x}{1 + wj}\right) \]

      8. fma-udef 92.0%

         \[\leadsto wj - \left(-\frac{\color{blue}{\mathsf{fma}\left(-1, wj, 1\right)} \cdot x}{1 + wj}\right) \]

      9. +-commutative 92.0%

         \[\leadsto wj - \left(-\frac{\mathsf{fma}\left(-1, wj, 1\right) \cdot x}{\color{blue}{wj + 1}}\right) \]

      10. associate-/l* 91.6%

          \[\leadsto wj - \left(-\color{blue}{\frac{\mathsf{fma}\left(-1, wj, 1\right)}{\frac{wj + 1}{x}}}\right) \]

      11. fma-udef 91.6%

          \[\leadsto wj - \left(-\frac{\color{blue}{-1 \cdot wj + 1}}{\frac{wj + 1}{x}}\right) \]

      12. neg-mul-1 91.6%

          \[\leadsto wj - \left(-\frac{\color{blue}{\left(-wj\right)} + 1}{\frac{wj + 1}{x}}\right) \]

      13. +-commutative 91.6%

          \[\leadsto wj - \left(-\frac{\color{blue}{1 + \left(-wj\right)}}{\frac{wj + 1}{x}}\right) \]

      14. sub-neg 91.6%

          \[\leadsto wj - \left(-\frac{\color{blue}{1 - wj}}{\frac{wj + 1}{x}}\right) \]

   10. Simplified 91.6%

       \[\leadsto wj - \color{blue}{\left(-\frac{1 - wj}{\frac{wj + 1}{x}}\right)} \]

   if -3.6999999999999998e-152 < x < 6.19999999999999999e-230

   1. Initial program 31.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 33.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 33.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 31.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 35.6%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 35.6%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 35.6%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 21.3%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 21.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 21.3%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in wj around inf 63.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{wj} + \frac{1}{{wj}^{2}}}} \]
   8. Step-by-step derivation

      1. inv-pow 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{\left({wj}^{2}\right)}^{-1}}} \]

      2. unpow2 63.9%

         \[\leadsto \frac{1}{\frac{1}{wj} + {\color{blue}{\left(wj \cdot wj\right)}}^{-1}} \]

      3. pow-prod-down 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{{wj}^{-1} \cdot {wj}^{-1}}} \]

      4. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj}} \cdot {wj}^{-1}} \]

      5. inv-pow 63.5%

         \[\leadsto \frac{1}{\frac{1}{wj} + \frac{1}{wj} \cdot \color{blue}{\frac{1}{wj}}} \]

   9. Applied egg-rr 63.5%

      \[\leadsto \frac{1}{\frac{1}{wj} + \color{blue}{\frac{1}{wj} \cdot \frac{1}{wj}}} \]
   10. Step-by-step derivation

       1. inv-pow 63.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{wj} + \frac{1}{wj} \cdot \frac{1}{wj}\right)}^{-1}} \]

       2. distribute-rgt1-in 63.5%

          \[\leadsto {\color{blue}{\left(\left(\frac{1}{wj} + 1\right) \cdot \frac{1}{wj}\right)}}^{-1} \]

       3. unpow-prod-down 63.7%

          \[\leadsto \color{blue}{{\left(\frac{1}{wj} + 1\right)}^{-1} \cdot {\left(\frac{1}{wj}\right)}^{-1}} \]

       4. +-commutative 63.7%

          \[\leadsto {\color{blue}{\left(1 + \frac{1}{wj}\right)}}^{-1} \cdot {\left(\frac{1}{wj}\right)}^{-1} \]

   11. Applied egg-rr 63.7%

       \[\leadsto \color{blue}{{\left(1 + \frac{1}{wj}\right)}^{-1} \cdot {\left(\frac{1}{wj}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 63.7%

          \[\leadsto {\left(1 + \frac{1}{wj}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{1}{wj}}} \]

       2. remove-double-div 63.8%

          \[\leadsto {\left(1 + \frac{1}{wj}\right)}^{-1} \cdot \color{blue}{wj} \]

       3. *-commutative 63.8%

          \[\leadsto \color{blue}{wj \cdot {\left(1 + \frac{1}{wj}\right)}^{-1}} \]

       4. unpow-1 63.8%

          \[\leadsto wj \cdot \color{blue}{\frac{1}{1 + \frac{1}{wj}}} \]

   13. Simplified 63.8%

       \[\leadsto \color{blue}{wj \cdot \frac{1}{1 + \frac{1}{wj}}} \]

   if 6.19999999999999999e-230 < x

   1. Initial program 81.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 81.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 81.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 81.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 81.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 81.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 81.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 80.8%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + -1 \cdot \left(wj \cdot x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 80.8%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-1 \cdot wj\right) \cdot x}\right)}{wj + 1} \]

      2. neg-mul-1 80.8%

         \[\leadsto wj - \frac{wj - \left(x + \color{blue}{\left(-wj\right)} \cdot x\right)}{wj + 1} \]

   7. Simplified 80.8%

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

      1. clear-num 80.6%

         \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \left(x + \left(-wj\right) \cdot x\right)}}} \]

      2. inv-pow 80.6%

         \[\leadsto wj - \color{blue}{{\left(\frac{wj + 1}{wj - \left(x + \left(-wj\right) \cdot x\right)}\right)}^{-1}} \]

      3. distribute-rgt1-in 80.6%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \color{blue}{\left(\left(-wj\right) + 1\right) \cdot x}}\right)}^{-1} \]

      4. neg-mul-1 80.6%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \left(\color{blue}{-1 \cdot wj} + 1\right) \cdot x}\right)}^{-1} \]

      5. fma-def 80.6%

         \[\leadsto wj - {\left(\frac{wj + 1}{wj - \color{blue}{\mathsf{fma}\left(-1, wj, 1\right)} \cdot x}\right)}^{-1} \]

   9. Applied egg-rr 80.6%

      \[\leadsto wj - \color{blue}{{\left(\frac{wj + 1}{wj - \mathsf{fma}\left(-1, wj, 1\right) \cdot x}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 80.6%

          \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - \mathsf{fma}\left(-1, wj, 1\right) \cdot x}}} \]

       2. fma-udef 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(-1 \cdot wj + 1\right)} \cdot x}} \]

       3. neg-mul-1 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \left(\color{blue}{\left(-wj\right)} + 1\right) \cdot x}} \]

       4. +-commutative 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x}} \]

       5. sub-neg 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{\left(1 - wj\right)} \cdot x}} \]

       6. *-commutative 80.6%

          \[\leadsto wj - \frac{1}{\frac{wj + 1}{wj - \color{blue}{x \cdot \left(1 - wj\right)}}} \]

   11. Simplified 80.6%

       \[\leadsto wj - \color{blue}{\frac{1}{\frac{wj + 1}{wj - x \cdot \left(1 - wj\right)}}} \]

   12. Taylor expanded in x around -inf 90.7%

       \[\leadsto \color{blue}{\frac{x \cdot \left(1 - wj\right)}{1 + wj}} \]
   13. Step-by-step derivation

       1. associate-/l* 90.7%

          \[\leadsto \color{blue}{\frac{x}{\frac{1 + wj}{1 - wj}}} \]

       2. +-commutative 90.7%

          \[\leadsto \frac{x}{\frac{\color{blue}{wj + 1}}{1 - wj}} \]

   14. Simplified 90.7%

       \[\leadsto \color{blue}{\frac{x}{\frac{wj + 1}{1 - wj}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-152}:\\ \;\;\;\;wj + \frac{1 - wj}{\frac{wj + 1}{x}}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-230}:\\ \;\;\;\;wj \cdot \frac{1}{1 + \frac{1}{wj}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 82.6% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-203} \lor \neg \left(x \leq 5.6 \cdot 10^{-247}\right):\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -1.05e-203) (not (<= x 5.6e-247)))
   (+ x (* -2.0 (* wj x)))
   (* wj (+ wj (* x -2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -1.05e-203) || !(x <= 5.6e-247)) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d-203)) .or. (.not. (x <= 5.6d-247))) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj * (wj + (x * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -1.05e-203) || !(x <= 5.6e-247)) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -1.05e-203) or not (x <= 5.6e-247):
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj * (wj + (x * -2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -1.05e-203) || !(x <= 5.6e-247))
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -1.05e-203) || ~((x <= 5.6e-247)))
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj * (wj + (x * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -1.05e-203], N[Not[LessEqual[x, 5.6e-247]], $MachinePrecision]], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000001e-203 or 5.59999999999999973e-247 < x

   1. Initial program 83.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 83.8%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 83.8%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 83.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 83.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 86.1%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 86.1%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 86.1%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 87.4%

      \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 87.4%

         \[\leadsto x + -2 \cdot \color{blue}{\left(x \cdot wj\right)} \]

   7. Simplified 87.4%

      \[\leadsto \color{blue}{x + -2 \cdot \left(x \cdot wj\right)} \]

   if -1.05000000000000001e-203 < x < 5.59999999999999973e-247

   1. Initial program 24.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 27.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 27.6%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 24.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 24.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 27.6%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 27.6%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 27.6%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 87.5%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]

   6. Taylor expanded in x around 0 87.5%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \color{blue}{{wj}^{2}}\right) \]

   7. Taylor expanded in wj around inf 60.0%

      \[\leadsto \color{blue}{-2 \cdot \left(wj \cdot x\right) + {wj}^{2}} \]
   8. Step-by-step derivation

      1. +-commutative 60.0%

         \[\leadsto \color{blue}{{wj}^{2} + -2 \cdot \left(wj \cdot x\right)} \]

      2. unpow2 60.0%

         \[\leadsto \color{blue}{wj \cdot wj} + -2 \cdot \left(wj \cdot x\right) \]

      3. *-commutative 60.0%

         \[\leadsto wj \cdot wj + \color{blue}{\left(wj \cdot x\right) \cdot -2} \]

      4. associate-*r* 60.0%

         \[\leadsto wj \cdot wj + \color{blue}{wj \cdot \left(x \cdot -2\right)} \]

      5. distribute-lft-out 60.0%

         \[\leadsto \color{blue}{wj \cdot \left(wj + x \cdot -2\right)} \]

      6. *-commutative 60.0%

         \[\leadsto wj \cdot \left(wj + \color{blue}{-2 \cdot x}\right) \]

   9. Simplified 60.0%

      \[\leadsto \color{blue}{wj \cdot \left(wj + -2 \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-203} \lor \neg \left(x \leq 5.6 \cdot 10^{-247}\right):\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 82.2% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-205}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-248}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -9.5e-205) x (if (<= x 3.1e-248) (* wj (+ wj (* x -2.0))) x)))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -9.5e-205) {
		tmp = x;
	} else if (x <= 3.1e-248) {
		tmp = wj * (wj + (x * -2.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-9.5d-205)) then
        tmp = x
    else if (x <= 3.1d-248) then
        tmp = wj * (wj + (x * (-2.0d0)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -9.5e-205) {
		tmp = x;
	} else if (x <= 3.1e-248) {
		tmp = wj * (wj + (x * -2.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -9.5e-205:
		tmp = x
	elif x <= 3.1e-248:
		tmp = wj * (wj + (x * -2.0))
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -9.5e-205)
		tmp = x;
	elseif (x <= 3.1e-248)
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -9.5e-205)
		tmp = x;
	elseif (x <= 3.1e-248)
		tmp = wj * (wj + (x * -2.0));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -9.5e-205], x, If[LessEqual[x, 3.1e-248], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999957e-205 or 3.1000000000000002e-248 < x

   1. Initial program 83.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 83.8%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 83.8%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 83.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 83.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 86.1%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 86.1%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 86.1%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 85.8%

      \[\leadsto \color{blue}{x} \]

   if -9.49999999999999957e-205 < x < 3.1000000000000002e-248

   1. Initial program 24.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 27.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 27.6%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 24.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 24.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 27.6%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 27.6%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 27.6%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 87.5%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]

   6. Taylor expanded in x around 0 87.5%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \color{blue}{{wj}^{2}}\right) \]

   7. Taylor expanded in wj around inf 60.0%

      \[\leadsto \color{blue}{-2 \cdot \left(wj \cdot x\right) + {wj}^{2}} \]
   8. Step-by-step derivation

      1. +-commutative 60.0%

         \[\leadsto \color{blue}{{wj}^{2} + -2 \cdot \left(wj \cdot x\right)} \]

      2. unpow2 60.0%

         \[\leadsto \color{blue}{wj \cdot wj} + -2 \cdot \left(wj \cdot x\right) \]

      3. *-commutative 60.0%

         \[\leadsto wj \cdot wj + \color{blue}{\left(wj \cdot x\right) \cdot -2} \]

      4. associate-*r* 60.0%

         \[\leadsto wj \cdot wj + \color{blue}{wj \cdot \left(x \cdot -2\right)} \]

      5. distribute-lft-out 60.0%

         \[\leadsto \color{blue}{wj \cdot \left(wj + x \cdot -2\right)} \]

      6. *-commutative 60.0%

         \[\leadsto wj \cdot \left(wj + \color{blue}{-2 \cdot x}\right) \]

   9. Simplified 60.0%

      \[\leadsto \color{blue}{wj \cdot \left(wj + -2 \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-205}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-248}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 85.8% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.75e-6) (+ x (* -2.0 (* wj x))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.75e-6) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.75d-6) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.75e-6) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.75e-6:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.75e-6)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.75e-6)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.75e-6], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.74999999999999997e-6

   1. Initial program 77.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 77.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 77.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 77.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 77.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 77.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 77.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 81.9%

      \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 81.9%

         \[\leadsto x + -2 \cdot \color{blue}{\left(x \cdot wj\right)} \]

   7. Simplified 81.9%

      \[\leadsto \color{blue}{x + -2 \cdot \left(x \cdot wj\right)} \]

   if 1.74999999999999997e-6 < wj

   1. Initial program 23.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 23.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 23.9%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   6. Step-by-step derivation

      1. +-commutative 95.3%

         \[\leadsto wj - \frac{wj}{\color{blue}{wj + 1}} \]

   7. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj}{wj + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 85.9% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.1e-6) (/ x (+ 1.0 (* wj 2.0))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.1e-6) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.1d-6) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.1e-6) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.1e-6:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.1e-6)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.1e-6)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.1e-6], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.0999999999999998e-6

   1. Initial program 77.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 77.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 77.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 77.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 77.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 77.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 77.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 52.8%

         \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      2. clear-num 52.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{wj \cdot wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}}} \]

      3. pow2 52.7%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{\color{blue}{{wj}^{2}} - \frac{wj - \frac{x}{e^{wj}}}{wj + 1} \cdot \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}} \]

      4. pow2 52.7%

         \[\leadsto \frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - \color{blue}{{\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   6. Applied egg-rr 52.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{wj + \frac{wj - \frac{x}{e^{wj}}}{wj + 1}}{{wj}^{2} - {\left(\frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{2}}}} \]

   7. Taylor expanded in x around inf 83.2%

      \[\leadsto \color{blue}{\frac{x}{e^{wj} \cdot \left(1 + wj\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 83.2%

         \[\leadsto \frac{x}{\color{blue}{\left(1 + wj\right) \cdot e^{wj}}} \]

      2. +-commutative 83.2%

         \[\leadsto \frac{x}{\color{blue}{\left(wj + 1\right)} \cdot e^{wj}} \]

   9. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{x}{\left(wj + 1\right) \cdot e^{wj}}} \]

   10. Taylor expanded in wj around 0 81.9%

       \[\leadsto \frac{x}{\color{blue}{1 + 2 \cdot wj}} \]
   11. Step-by-step derivation

       1. *-commutative 81.9%

          \[\leadsto \frac{x}{1 + \color{blue}{wj \cdot 2}} \]

   12. Simplified 81.9%

       \[\leadsto \frac{x}{\color{blue}{1 + wj \cdot 2}} \]

   if 2.0999999999999998e-6 < wj

   1. Initial program 23.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 23.5%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 23.9%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 23.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 95.3%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 95.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   6. Step-by-step derivation

      1. +-commutative 95.3%

         \[\leadsto wj - \frac{wj}{\color{blue}{wj + 1}} \]

   7. Simplified 95.3%

      \[\leadsto \color{blue}{wj - \frac{wj}{wj + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 84.9% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.75:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= wj 1.75) x (+ wj -1.0)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.75) {
		tmp = x;
	} else {
		tmp = wj + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.75d0) then
        tmp = x
    else
        tmp = wj + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.75) {
		tmp = x;
	} else {
		tmp = wj + -1.0;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.75:
		tmp = x
	else:
		tmp = wj + -1.0
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.75)
		tmp = x;
	else
		tmp = Float64(wj + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.75)
		tmp = x;
	else
		tmp = wj + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.75], x, N[(wj + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.75

   1. Initial program 77.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 77.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 77.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 77.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 77.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 77.4%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 77.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 80.0%

      \[\leadsto \color{blue}{x} \]

   if 1.75 < wj

   1. Initial program 0.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 0.0%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. associate-/l/ 0.0%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      3. div-sub 0.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      4. associate-/l* 0.0%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      5. *-inverses 100.0%

         \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. /-rgt-identity 100.0%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around inf 87.3%

      \[\leadsto \color{blue}{wj - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.75:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 4.3% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ wj \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 wj)

C

double code(double wj, double x) {
	return wj;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj
end function

Java

public static double code(double wj, double x) {
	return wj;
}

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

function tmp = code(wj, x)
	tmp = wj;
end

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 75.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. distribute-rgt1-in 75.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

   2. associate-/l/ 75.9%

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

   3. div-sub 75.5%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

   4. associate-/l* 75.5%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

   5. *-inverses 77.9%

      \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

   6. /-rgt-identity 77.9%

      \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

3. Simplified 77.9%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around inf 4.6%

   \[\leadsto \color{blue}{wj} \]

6. Final simplification 4.6%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 22: 84.2% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 x)

C

double code(double wj, double x) {
	return x;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double wj, double x) {
	return x;
}

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

function tmp = code(wj, x)
	tmp = x;
end

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 75.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. distribute-rgt1-in 75.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

   2. associate-/l/ 75.9%

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

   3. div-sub 75.5%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

   4. associate-/l* 75.5%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj}{\frac{e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj}}}{wj + 1} \]

   5. *-inverses 77.9%

      \[\leadsto wj - \frac{\frac{wj}{\color{blue}{1}} - \frac{x}{e^{wj}}}{wj + 1} \]

   6. /-rgt-identity 77.9%

      \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

3. Simplified 77.9%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 78.4%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 78.4%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 77.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj)))))))

C

double code(double wj, double x) {
	return wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj - ((wj / (wj + 1.0d0)) - (x / (exp(wj) + (wj * exp(wj)))))
end function

Java

public static double code(double wj, double x) {
	return wj - ((wj / (wj + 1.0)) - (x / (Math.exp(wj) + (wj * Math.exp(wj)))));
}

Python

def code(wj, x):
	return wj - ((wj / (wj + 1.0)) - (x / (math.exp(wj) + (wj * math.exp(wj)))))

Julia

function code(wj, x)
	return Float64(wj - Float64(Float64(wj / Float64(wj + 1.0)) - Float64(x / Float64(exp(wj) + Float64(wj * exp(wj))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
end

Wolfram

code[wj_, x_] := N[(wj - N[(N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Exp[wj], $MachinePrecision] + N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024010 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))