Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.7% → 99.0%

Time: 10.3s

Alternatives: 12

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -3.903909540158924e+77
  2. x = 1.8585672695054658e+93

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: 77.7% 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: 99.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj - \frac{x}{e^{wj}}\\ \mathbf{if}\;wj \leq -6.8 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{t\_0}{-1 - wj}\\ \mathbf{elif}\;wj \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + t\_0 \cdot \frac{-1}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- wj (/ x (exp wj)))))
   (if (<= wj -6.8e-9)
     (+ wj (/ t_0 (- -1.0 wj)))
     (if (<= wj 5.2e-19)
       (+ x (+ (* -2.0 (* wj x)) (pow wj 2.0)))
       (+ wj (* t_0 (/ -1.0 (+ wj 1.0))))))))

C

double code(double wj, double x) {
	double t_0 = wj - (x / exp(wj));
	double tmp;
	if (wj <= -6.8e-9) {
		tmp = wj + (t_0 / (-1.0 - wj));
	} else if (wj <= 5.2e-19) {
		tmp = x + ((-2.0 * (wj * x)) + pow(wj, 2.0));
	} else {
		tmp = wj + (t_0 * (-1.0 / (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) :: tmp
    t_0 = wj - (x / exp(wj))
    if (wj <= (-6.8d-9)) then
        tmp = wj + (t_0 / ((-1.0d0) - wj))
    else if (wj <= 5.2d-19) then
        tmp = x + (((-2.0d0) * (wj * x)) + (wj ** 2.0d0))
    else
        tmp = wj + (t_0 * ((-1.0d0) / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj - (x / Math.exp(wj));
	double tmp;
	if (wj <= -6.8e-9) {
		tmp = wj + (t_0 / (-1.0 - wj));
	} else if (wj <= 5.2e-19) {
		tmp = x + ((-2.0 * (wj * x)) + Math.pow(wj, 2.0));
	} else {
		tmp = wj + (t_0 * (-1.0 / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj - (x / math.exp(wj))
	tmp = 0
	if wj <= -6.8e-9:
		tmp = wj + (t_0 / (-1.0 - wj))
	elif wj <= 5.2e-19:
		tmp = x + ((-2.0 * (wj * x)) + math.pow(wj, 2.0))
	else:
		tmp = wj + (t_0 * (-1.0 / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj - Float64(x / exp(wj)))
	tmp = 0.0
	if (wj <= -6.8e-9)
		tmp = Float64(wj + Float64(t_0 / Float64(-1.0 - wj)));
	elseif (wj <= 5.2e-19)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + (wj ^ 2.0)));
	else
		tmp = Float64(wj + Float64(t_0 * Float64(-1.0 / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj - (x / exp(wj));
	tmp = 0.0;
	if (wj <= -6.8e-9)
		tmp = wj + (t_0 / (-1.0 - wj));
	elseif (wj <= 5.2e-19)
		tmp = x + ((-2.0 * (wj * x)) + (wj ^ 2.0));
	else
		tmp = wj + (t_0 * (-1.0 / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -6.8e-9], N[(wj + N[(t$95$0 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5.2e-19], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(t$95$0 * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -6.7999999999999997e-9

   1. Initial program 95.6%

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

      1. distribute-rgt1-in 95.8%

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

      2. associate-/l/ 95.8%

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

      3. div-sub 95.8%

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

      4. associate-/l* 95.8%

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

      5. *-inverses 95.8%

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

      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 -6.7999999999999997e-9 < wj < 5.20000000000000026e-19

   1. Initial program 82.5%

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

      1. distribute-rgt1-in 82.5%

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

      2. associate-/l/ 82.5%

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

      3. div-sub 82.5%

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

      4. associate-/l* 82.5%

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

      5. *-inverses 82.5%

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

      6. *-rgt-identity 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in wj around 0 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

   if 5.20000000000000026e-19 < wj

   1. Initial program 69.5%

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

      1. distribute-rgt1-in 69.4%

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

      2. associate-/l/ 69.8%

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

      3. div-sub 69.8%

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

      4. associate-/l* 69.8%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.8 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ t_1 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_1}{e^{wj} + t\_1} \leq 10^{-19}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - t\_0\right) - {wj}^{3} \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj - \frac{x}{e^{wj}}, \frac{-1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))) (t_1 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_1) (+ (exp wj) t_1))) 1e-19)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (-
        (* (pow wj 2.0) (- 1.0 t_0))
        (*
         (pow wj 3.0)
         (+ 1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))))
     (fma (- wj (/ x (exp wj))) (/ -1.0 (+ wj 1.0)) wj))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double t_1 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 1e-19) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 2.0) * (1.0 - t_0)) - (pow(wj, 3.0) * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))));
	} else {
		tmp = fma((wj - (x / exp(wj))), (-1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	t_1 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_1) / Float64(exp(wj) + t_1))) <= 1e-19)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 2.0) * Float64(1.0 - t_0)) - Float64((wj ^ 3.0) * Float64(1.0 + Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666))))))));
	else
		tmp = fma(Float64(wj - Float64(x / exp(wj))), Float64(-1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$1), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-19], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[Power[wj, 3.0], $MachinePrecision] * N[(1.0 + N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $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))))) < 9.9999999999999998e-20

   1. Initial program 75.4%

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

      1. distribute-rgt1-in 75.4%

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

      2. associate-/l/ 75.4%

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

      3. div-sub 75.4%

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

      4. associate-/l* 75.4%

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

      5. *-inverses 75.4%

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

      6. *-rgt-identity 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in wj around 0 98.5%

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

   if 9.9999999999999998e-20 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.1%

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

      1. distribute-rgt1-in 96.1%

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

      2. associate-/l/ 96.2%

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

      3. div-sub 96.2%

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

      4. associate-/l* 96.2%

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

      5. *-inverses 99.7%

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

      6. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. div-inv 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

      5. fma-define 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-19}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3} \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj - \frac{x}{e^{wj}}, \frac{-1}{wj + 1}, wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.2e-19)
   (+
    x
    (+
     (* -2.0 (* wj x))
     (- (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5)))) (pow wj 3.0))))
   (+ wj (* (- wj (/ x (exp wj))) (/ -1.0 (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.2e-19) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - pow(wj, 3.0)));
	} else {
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (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 <= 5.2d-19) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))) - (wj ** 3.0d0)))
    else
        tmp = wj + ((wj - (x / exp(wj))) * ((-1.0d0) / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 5.2e-19) {
		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 = wj + ((wj - (x / Math.exp(wj))) * (-1.0 / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.2e-19:
		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 = wj + ((wj - (x / math.exp(wj))) * (-1.0 / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.2e-19)
		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(wj + Float64(Float64(wj - Float64(x / exp(wj))) * Float64(-1.0 / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.2e-19)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - (wj ^ 3.0)));
	else
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.2e-19], 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[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.20000000000000026e-19

   1. Initial program 82.8%

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

      1. distribute-rgt1-in 82.8%

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

      2. associate-/l/ 82.8%

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

      3. div-sub 82.8%

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

      4. associate-/l* 82.8%

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

      5. *-inverses 82.8%

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

      6. *-rgt-identity 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in wj around 0 98.4%

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

   6. Taylor expanded in x around 0 98.4%

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

   if 5.20000000000000026e-19 < wj

   1. Initial program 69.5%

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

      1. distribute-rgt1-in 69.4%

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

      2. associate-/l/ 69.8%

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

      3. div-sub 69.8%

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

      4. associate-/l* 69.8%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.2e-19)
   (+
    x
    (+ (* -2.0 (* wj x)) (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5))))))
   (+ wj (* (- wj (/ x (exp wj))) (/ -1.0 (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.2e-19) {
		tmp = x + ((-2.0 * (wj * x)) + (pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	} else {
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (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 <= 5.2d-19) then
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))))
    else
        tmp = wj + ((wj - (x / exp(wj))) * ((-1.0d0) / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 5.2e-19) {
		tmp = x + ((-2.0 * (wj * x)) + (Math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	} else {
		tmp = wj + ((wj - (x / Math.exp(wj))) * (-1.0 / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.2e-19:
		tmp = x + ((-2.0 * (wj * x)) + (math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))))
	else:
		tmp = wj + ((wj - (x / math.exp(wj))) * (-1.0 / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.2e-19)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64((wj ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5))))));
	else
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) * Float64(-1.0 / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.2e-19)
		tmp = x + ((-2.0 * (wj * x)) + ((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	else
		tmp = wj + ((wj - (x / exp(wj))) * (-1.0 / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.2e-19], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.20000000000000026e-19

   1. Initial program 82.8%

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

      1. distribute-rgt1-in 82.8%

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

      2. associate-/l/ 82.8%

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

      3. div-sub 82.8%

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

      4. associate-/l* 82.8%

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

      5. *-inverses 82.8%

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

      6. *-rgt-identity 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in wj around 0 98.2%

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

   if 5.20000000000000026e-19 < wj

   1. Initial program 69.5%

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

      1. distribute-rgt1-in 69.4%

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

      2. associate-/l/ 69.8%

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

      3. div-sub 69.8%

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

      4. associate-/l* 69.8%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6.8 \cdot 10^{-9} \lor \neg \left(wj \leq 5.2 \cdot 10^{-19}\right):\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -6.8e-9) (not (<= wj 5.2e-19)))
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
   (+ x (+ (* -2.0 (* wj x)) (pow wj 2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -6.8e-9) || !(wj <= 5.2e-19)) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} 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 <= (-6.8d-9)) .or. (.not. (wj <= 5.2d-19))) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    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 <= -6.8e-9) || !(wj <= 5.2e-19)) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + Math.pow(wj, 2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -6.8e-9) or not (wj <= 5.2e-19):
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	else:
		tmp = x + ((-2.0 * (wj * x)) + math.pow(wj, 2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -6.8e-9) || !(wj <= 5.2e-19))
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	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 <= -6.8e-9) || ~((wj <= 5.2e-19)))
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	else
		tmp = x + ((-2.0 * (wj * x)) + (wj ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -6.8e-9], N[Not[LessEqual[wj, 5.2e-19]], $MachinePrecision]], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $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 < -6.7999999999999997e-9 or 5.20000000000000026e-19 < wj

   1. Initial program 79.3%

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

      1. distribute-rgt1-in 79.3%

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

      2. associate-/l/ 79.6%

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

      3. div-sub 79.6%

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

      4. associate-/l* 79.6%

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

      5. *-inverses 98.3%

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

      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

   if -6.7999999999999997e-9 < wj < 5.20000000000000026e-19

   1. Initial program 82.5%

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

      1. distribute-rgt1-in 82.5%

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

      2. associate-/l/ 82.5%

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

      3. div-sub 82.5%

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

      4. associate-/l* 82.5%

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

      5. *-inverses 82.5%

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

      6. *-rgt-identity 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in wj around 0 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.8 \cdot 10^{-9} \lor \neg \left(wj \leq 5.2 \cdot 10^{-19}\right):\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (+ (* -2.0 (* wj x)) (pow wj 2.0))))

C

double code(double wj, double x) {
	return x + ((-2.0 * (wj * x)) + pow(wj, 2.0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (((-2.0d0) * (wj * x)) + (wj ** 2.0d0))
end function

Java

public static double code(double wj, double x) {
	return x + ((-2.0 * (wj * x)) + Math.pow(wj, 2.0));
}

Python

def code(wj, x):
	return x + ((-2.0 * (wj * x)) + math.pow(wj, 2.0))

Julia

function code(wj, x)
	return Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + (wj ^ 2.0)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + ((-2.0 * (wj * x)) + (wj ^ 2.0));
end

Wolfram

code[wj_, x_] := N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

   1. distribute-rgt1-in 82.3%

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

   2. associate-/l/ 82.3%

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

   3. div-sub 82.3%

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

   4. associate-/l* 82.3%

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

   5. *-inverses 83.5%

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

   6. *-rgt-identity 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in wj around 0 96.5%

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

6. Taylor expanded in x around 0 96.1%

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

7. Final simplification 96.1%

   \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right) \]
8. Add Preprocessing

Alternative 7: 85.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{e^{wj} \cdot \left(wj + 1\right)} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (/ x (* (exp wj) (+ wj 1.0))))

C

double code(double wj, double x) {
	return x / (exp(wj) * (wj + 1.0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x / (exp(wj) * (wj + 1.0d0))
end function

Java

public static double code(double wj, double x) {
	return x / (Math.exp(wj) * (wj + 1.0));
}

Python

def code(wj, x):
	return x / (math.exp(wj) * (wj + 1.0))

Julia

function code(wj, x)
	return Float64(x / Float64(exp(wj) * Float64(wj + 1.0)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x / (exp(wj) * (wj + 1.0));
end

Wolfram

code[wj_, x_] := N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

   1. distribute-rgt1-in 82.3%

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

   2. associate-/l/ 82.3%

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

   3. div-sub 82.3%

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

   4. associate-/l* 82.3%

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

   5. *-inverses 83.5%

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

   6. *-rgt-identity 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in x around inf 90.4%

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

   1. +-commutative 90.4%

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

7. Simplified 90.4%

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

8. Final simplification 90.4%

   \[\leadsto \frac{x}{e^{wj} \cdot \left(wj + 1\right)} \]
9. Add Preprocessing

Alternative 8: 85.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{e^{wj}}}{wj + 1} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (/ (/ x (exp wj)) (+ wj 1.0)))

C

double code(double wj, double x) {
	return (x / exp(wj)) / (wj + 1.0);
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = (x / exp(wj)) / (wj + 1.0d0)
end function

Java

public static double code(double wj, double x) {
	return (x / Math.exp(wj)) / (wj + 1.0);
}

Python

def code(wj, x):
	return (x / math.exp(wj)) / (wj + 1.0)

Julia

function code(wj, x)
	return Float64(Float64(x / exp(wj)) / Float64(wj + 1.0))
end

MATLAB

function tmp = code(wj, x)
	tmp = (x / exp(wj)) / (wj + 1.0);
end

Wolfram

code[wj_, x_] := N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

   1. distribute-rgt1-in 82.3%

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

   2. associate-/l/ 82.3%

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

   3. div-sub 82.3%

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

   4. associate-/l* 82.3%

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

   5. *-inverses 83.5%

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

   6. *-rgt-identity 83.5%

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

3. Simplified 83.5%

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

   1. clear-num 83.3%

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

   2. associate-/r/ 83.5%

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

   3. rec-exp 83.4%

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

6. Applied egg-rr 83.4%

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

7. Taylor expanded in x around inf 90.4%

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

   1. exp-neg 90.4%

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

   2. associate-*r/ 90.4%

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

   3. *-rgt-identity 90.4%

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

9. Simplified 90.4%

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

10. Final simplification 90.4%

    \[\leadsto \frac{\frac{x}{e^{wj}}}{wj + 1} \]
11. Add Preprocessing

Alternative 9: 84.2% accurate, 44.7× speedup

Math

\[\begin{array}{l} \\ x + -2 \cdot \left(wj \cdot x\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (* -2.0 (* wj x))))

C

double code(double wj, double x) {
	return x + (-2.0 * (wj * x));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + ((-2.0d0) * (wj * x))
end function

Java

public static double code(double wj, double x) {
	return x + (-2.0 * (wj * x));
}

Python

def code(wj, x):
	return x + (-2.0 * (wj * x))

Julia

function code(wj, x)
	return Float64(x + Float64(-2.0 * Float64(wj * x)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (-2.0 * (wj * x));
end

Wolfram

code[wj_, x_] := N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

   1. distribute-rgt1-in 82.3%

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

   2. associate-/l/ 82.3%

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

   3. div-sub 82.3%

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

   4. associate-/l* 82.3%

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

   5. *-inverses 83.5%

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

   6. *-rgt-identity 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in wj around 0 88.5%

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

   1. *-commutative 88.5%

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

7. Simplified 88.5%

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

8. Final simplification 88.5%

   \[\leadsto x + -2 \cdot \left(wj \cdot x\right) \]
9. Add Preprocessing

Alternative 10: 84.3% accurate, 44.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + wj \cdot 2} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (/ x (+ 1.0 (* wj 2.0))))

C

double code(double wj, double x) {
	return x / (1.0 + (wj * 2.0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x / (1.0d0 + (wj * 2.0d0))
end function

Java

public static double code(double wj, double x) {
	return x / (1.0 + (wj * 2.0));
}

Python

def code(wj, x):
	return x / (1.0 + (wj * 2.0))

Julia

function code(wj, x)
	return Float64(x / Float64(1.0 + Float64(wj * 2.0)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x / (1.0 + (wj * 2.0));
end

Wolfram

code[wj_, x_] := N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

   1. distribute-rgt1-in 82.3%

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

   2. associate-/l/ 82.3%

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

   3. div-sub 82.3%

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

   4. associate-/l* 82.3%

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

   5. *-inverses 83.5%

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

   6. *-rgt-identity 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in x around inf 90.4%

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

   1. +-commutative 90.4%

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

7. Simplified 90.4%

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

8. Taylor expanded in wj around 0 88.5%

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

   1. *-commutative 88.5%

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

10. Simplified 88.5%

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

11. Final simplification 88.5%

    \[\leadsto \frac{x}{1 + wj \cdot 2} \]
12. Add Preprocessing

Alternative 11: 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 82.3%

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

   1. distribute-rgt1-in 82.3%

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

   2. associate-/l/ 82.3%

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

   3. div-sub 82.3%

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

   4. associate-/l* 82.3%

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

   5. *-inverses 83.5%

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

   6. *-rgt-identity 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in wj around inf 4.2%

   \[\leadsto \color{blue}{wj} \]

6. Final simplification 4.2%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 12: 83.7% 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 82.3%

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

   1. distribute-rgt1-in 82.3%

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

   2. associate-/l/ 82.3%

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

   3. div-sub 82.3%

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

   4. associate-/l* 82.3%

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

   5. *-inverses 83.5%

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

   6. *-rgt-identity 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in wj around 0 87.9%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 87.9%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 78.6% 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 2024043 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))