Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.4% → 98.7%

Time: 10.2s

Alternatives: 12

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.2121488450901671e+296
  2. x = -4.472577386166422e-241

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: 78.4% 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: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-13)
     (- (+ (* wj wj) (+ x (* -2.0 (* wj x)))) (pow wj 3.0))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-13) {
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - 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) :: t_0
    real(8) :: tmp
    t_0 = wj * exp(wj)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5d-13) then
        tmp = ((wj * wj) + (x + ((-2.0d0) * (wj * x)))) - (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 t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 5e-13) {
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - Math.pow(wj, 3.0);
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 5e-13:
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - math.pow(wj, 3.0)
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 5e-13)
		tmp = Float64(Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x)))) - (wj ^ 3.0));
	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);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-13)
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - (wj ^ 3.0);
	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]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-13], N[(N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $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))))) < 4.9999999999999999e-13

   1. Initial program 74.7%

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

      1. sub-neg 74.7%

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

      2. div-sub 74.7%

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

      3. sub-neg 74.7%

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

      4. +-commutative 74.7%

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

      5. distribute-neg-in 74.7%

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

      6. remove-double-neg 74.7%

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

      7. sub-neg 74.7%

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

      8. div-sub 74.7%

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

      9. distribute-rgt1-in 74.7%

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

      10. associate-/l/ 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in wj around 0 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

   if 4.9999999999999999e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.8%

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

      1. sub-neg 94.8%

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

      2. div-sub 94.8%

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

      3. sub-neg 94.8%

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

      4. +-commutative 94.8%

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

      5. distribute-neg-in 94.8%

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

      6. remove-double-neg 94.8%

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

      7. sub-neg 94.8%

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

      8. div-sub 94.8%

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

      9. distribute-rgt1-in 96.4%

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

      10. associate-/l/ 96.4%

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

   3. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 2: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj}} - wj\right) \cdot \frac{1}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.5e-9)
   (+ (* wj wj) (- x (* wj (+ x x))))
   (+ wj (* (- (/ x (exp wj)) wj) (/ 1.0 (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.5e-9) {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	} else {
		tmp = wj + (((x / exp(wj)) - 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 <= 5.5d-9) then
        tmp = (wj * wj) + (x - (wj * (x + x)))
    else
        tmp = wj + (((x / exp(wj)) - 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 <= 5.5e-9) {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) * (1.0 / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.5e-9:
		tmp = (wj * wj) + (x - (wj * (x + x)))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) * (1.0 / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.5e-9)
		tmp = Float64(Float64(wj * wj) + Float64(x - Float64(wj * Float64(x + x))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) * Float64(1.0 / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.5e-9)
		tmp = (wj * wj) + (x - (wj * (x + x)));
	else
		tmp = wj + (((x / exp(wj)) - wj) * (1.0 / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.5e-9], N[(N[(wj * wj), $MachinePrecision] + N[(x - N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.4999999999999996e-9

   1. Initial program 79.9%

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

      1. sub-neg 79.9%

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

      2. div-sub 79.9%

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

      3. sub-neg 79.9%

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

      4. +-commutative 79.9%

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

      5. distribute-neg-in 79.9%

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

      6. remove-double-neg 79.9%

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

      7. sub-neg 79.9%

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

      8. div-sub 79.9%

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

      9. distribute-rgt1-in 80.3%

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

      10. associate-/l/ 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in wj around 0 79.9%

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

      1. associate-*r* 79.9%

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

      2. neg-mul-1 79.9%

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

      3. distribute-lft1-in 79.9%

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

      4. +-commutative 79.9%

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

      5. sub-neg 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in wj around 0 99.1%

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

   8. Taylor expanded in x around 0 99.1%

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

      1. unpow2 99.1%

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

   10. Simplified 99.1%

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

   if 5.4999999999999996e-9 < wj

   1. Initial program 68.7%

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

      1. sub-neg 68.7%

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

      2. div-sub 68.7%

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

      3. sub-neg 68.7%

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

      4. +-commutative 68.7%

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

      5. distribute-neg-in 68.7%

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

      6. remove-double-neg 68.7%

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

      7. sub-neg 68.7%

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

      8. div-sub 68.7%

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

      9. distribute-rgt1-in 68.7%

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

      10. associate-/l/ 68.8%

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

   3. Simplified 97.4%

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

      1. clear-num 97.6%

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

      2. associate-/r/ 97.7%

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

   5. Applied egg-rr 97.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj}} - wj\right) \cdot \frac{1}{wj + 1}\\ \end{array} \]

Alternative 3: 97.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.5e-9)
   (+ (* wj wj) (- x (* wj (+ x x))))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.5e-9) {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	} 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 <= 5.5d-9) then
        tmp = (wj * wj) + (x - (wj * (x + x)))
    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 <= 5.5e-9) {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.5e-9:
		tmp = (wj * wj) + (x - (wj * (x + x)))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.5e-9)
		tmp = Float64(Float64(wj * wj) + Float64(x - Float64(wj * Float64(x + x))));
	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)
	tmp = 0.0;
	if (wj <= 5.5e-9)
		tmp = (wj * wj) + (x - (wj * (x + x)));
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.5e-9], N[(N[(wj * wj), $MachinePrecision] + N[(x - N[(wj * N[(x + x), $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 wj < 5.4999999999999996e-9

   1. Initial program 79.9%

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

      1. sub-neg 79.9%

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

      2. div-sub 79.9%

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

      3. sub-neg 79.9%

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

      4. +-commutative 79.9%

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

      5. distribute-neg-in 79.9%

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

      6. remove-double-neg 79.9%

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

      7. sub-neg 79.9%

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

      8. div-sub 79.9%

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

      9. distribute-rgt1-in 80.3%

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

      10. associate-/l/ 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in wj around 0 79.9%

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

      1. associate-*r* 79.9%

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

      2. neg-mul-1 79.9%

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

      3. distribute-lft1-in 79.9%

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

      4. +-commutative 79.9%

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

      5. sub-neg 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in wj around 0 99.1%

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

   8. Taylor expanded in x around 0 99.1%

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

      1. unpow2 99.1%

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

   10. Simplified 99.1%

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

   if 5.4999999999999996e-9 < wj

   1. Initial program 68.7%

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

      1. sub-neg 68.7%

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

      2. div-sub 68.7%

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

      3. sub-neg 68.7%

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

      4. +-commutative 68.7%

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

      5. distribute-neg-in 68.7%

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

      6. remove-double-neg 68.7%

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

      7. sub-neg 68.7%

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

      8. div-sub 68.7%

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

      9. distribute-rgt1-in 68.7%

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

      10. associate-/l/ 68.8%

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

   3. Simplified 97.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 4: 85.1% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{if}\;wj \leq -7.8 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq -6.5 \cdot 10^{-85}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(x + 1\right) - wj \cdot x\\ \mathbf{elif}\;wj \leq 0.0039:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;wj + wj \cdot \frac{-1}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ x (/ (+ wj 1.0) (- 1.0 wj)))))
   (if (<= wj -7.8e-62)
     t_0
     (if (<= wj -6.5e-85)
       (- (* (* wj wj) (+ x 1.0)) (* wj x))
       (if (<= wj 0.0039) t_0 (+ wj (* wj (/ -1.0 (+ wj 1.0)))))))))

C

double code(double wj, double x) {
	double t_0 = x / ((wj + 1.0) / (1.0 - wj));
	double tmp;
	if (wj <= -7.8e-62) {
		tmp = t_0;
	} else if (wj <= -6.5e-85) {
		tmp = ((wj * wj) * (x + 1.0)) - (wj * x);
	} else if (wj <= 0.0039) {
		tmp = t_0;
	} else {
		tmp = wj + (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) :: t_0
    real(8) :: tmp
    t_0 = x / ((wj + 1.0d0) / (1.0d0 - wj))
    if (wj <= (-7.8d-62)) then
        tmp = t_0
    else if (wj <= (-6.5d-85)) then
        tmp = ((wj * wj) * (x + 1.0d0)) - (wj * x)
    else if (wj <= 0.0039d0) then
        tmp = t_0
    else
        tmp = wj + (wj * ((-1.0d0) / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = x / ((wj + 1.0) / (1.0 - wj));
	double tmp;
	if (wj <= -7.8e-62) {
		tmp = t_0;
	} else if (wj <= -6.5e-85) {
		tmp = ((wj * wj) * (x + 1.0)) - (wj * x);
	} else if (wj <= 0.0039) {
		tmp = t_0;
	} else {
		tmp = wj + (wj * (-1.0 / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = x / ((wj + 1.0) / (1.0 - wj))
	tmp = 0
	if wj <= -7.8e-62:
		tmp = t_0
	elif wj <= -6.5e-85:
		tmp = ((wj * wj) * (x + 1.0)) - (wj * x)
	elif wj <= 0.0039:
		tmp = t_0
	else:
		tmp = wj + (wj * (-1.0 / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)))
	tmp = 0.0
	if (wj <= -7.8e-62)
		tmp = t_0;
	elseif (wj <= -6.5e-85)
		tmp = Float64(Float64(Float64(wj * wj) * Float64(x + 1.0)) - Float64(wj * x));
	elseif (wj <= 0.0039)
		tmp = t_0;
	else
		tmp = Float64(wj + Float64(wj * Float64(-1.0 / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = x / ((wj + 1.0) / (1.0 - wj));
	tmp = 0.0;
	if (wj <= -7.8e-62)
		tmp = t_0;
	elseif (wj <= -6.5e-85)
		tmp = ((wj * wj) * (x + 1.0)) - (wj * x);
	elseif (wj <= 0.0039)
		tmp = t_0;
	else
		tmp = wj + (wj * (-1.0 / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -7.8e-62], t$95$0, If[LessEqual[wj, -6.5e-85], N[(N[(N[(wj * wj), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(wj * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.0039], t$95$0, N[(wj + N[(wj * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -7.8000000000000007e-62 or -6.5e-85 < wj < 0.0038999999999999998

   1. Initial program 81.7%

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

      1. sub-neg 81.7%

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

      2. div-sub 81.7%

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

      3. sub-neg 81.7%

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

      4. +-commutative 81.7%

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

      5. distribute-neg-in 81.7%

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

      6. remove-double-neg 81.7%

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

      7. sub-neg 81.7%

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

      8. div-sub 81.7%

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

      9. distribute-rgt1-in 82.2%

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

      10. associate-/l/ 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in wj around 0 81.5%

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

      1. associate-*r* 81.5%

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

      2. neg-mul-1 81.5%

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

      3. distribute-lft1-in 81.5%

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

      4. +-commutative 81.5%

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

      5. sub-neg 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in x around -inf 90.7%

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

      1. *-commutative 90.7%

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

      2. associate-/l* 90.7%

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

      3. +-commutative 90.7%

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

   9. Simplified 90.7%

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

   if -7.8000000000000007e-62 < wj < -6.5e-85

   1. Initial program 18.3%

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

      1. sub-neg 18.3%

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

      2. div-sub 18.3%

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

      3. sub-neg 18.3%

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

      4. +-commutative 18.3%

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

      5. distribute-neg-in 18.3%

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

      6. remove-double-neg 18.3%

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

      7. sub-neg 18.3%

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

      8. div-sub 18.3%

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

      9. distribute-rgt1-in 18.3%

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

      10. associate-/l/ 18.3%

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

   3. Simplified 18.3%

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

      1. clear-num 18.0%

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

      2. associate-/r/ 18.3%

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

   5. Applied egg-rr 18.3%

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

   6. Taylor expanded in wj around 0 18.3%

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

      1. associate-*r* 18.3%

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

      2. neg-mul-1 18.3%

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

      3. distribute-lft1-in 18.3%

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

      4. +-commutative 18.3%

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

      5. sub-neg 18.3%

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

   8. Simplified 18.3%

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

   9. Taylor expanded in wj around inf 4.9%

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

       1. associate-*r* 4.9%

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

       2. mul-1-neg 4.9%

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

   11. Simplified 4.9%

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

   12. Taylor expanded in wj around 0 86.6%

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

       1. mul-1-neg 86.6%

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

       2. unsub-neg 86.6%

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

       3. +-commutative 86.6%

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

       4. unpow2 86.6%

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

       5. *-commutative 86.6%

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

   14. Simplified 86.6%

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

   if 0.0038999999999999998 < wj

   1. Initial program 63.8%

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

      1. sub-neg 63.8%

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

      2. div-sub 63.8%

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

      3. sub-neg 63.8%

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

      4. +-commutative 63.8%

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

      5. distribute-neg-in 63.8%

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

      6. remove-double-neg 63.8%

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

      7. sub-neg 63.8%

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

      8. div-sub 63.8%

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

      9. distribute-rgt1-in 63.8%

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

      10. associate-/l/ 63.9%

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

   3. Simplified 97.2%

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

      1. clear-num 97.2%

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

      2. associate-/r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in wj around 0 39.8%

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

      1. associate-*r* 39.5%

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

      2. neg-mul-1 39.5%

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

      3. distribute-lft1-in 39.5%

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

      4. +-commutative 39.5%

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

      5. sub-neg 39.5%

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

   8. Simplified 39.8%

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

   9. Taylor expanded in wj around inf 32.8%

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

       1. associate-*r* 32.8%

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

       2. mul-1-neg 32.8%

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

   11. Simplified 32.8%

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

   12. Taylor expanded in x around 0 81.9%

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

       1. mul-1-neg 81.9%

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

   14. Simplified 81.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -7.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{elif}\;wj \leq -6.5 \cdot 10^{-85}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(x + 1\right) - wj \cdot x\\ \mathbf{elif}\;wj \leq 0.0039:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj + wj \cdot \frac{-1}{wj + 1}\\ \end{array} \]

Alternative 5: 85.9% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;wj - \frac{wj - x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{elif}\;wj \leq -5.8 \cdot 10^{-85}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(x + 1\right) - wj \cdot x\\ \mathbf{elif}\;wj \leq 0.0039:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj + wj \cdot \frac{-1}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.45e-65)
   (- wj (/ (- wj (* x (- 1.0 wj))) (+ wj 1.0)))
   (if (<= wj -5.8e-85)
     (- (* (* wj wj) (+ x 1.0)) (* wj x))
     (if (<= wj 0.0039)
       (/ x (/ (+ wj 1.0) (- 1.0 wj)))
       (+ wj (* wj (/ -1.0 (+ wj 1.0))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.45e-65) {
		tmp = wj - ((wj - (x * (1.0 - wj))) / (wj + 1.0));
	} else if (wj <= -5.8e-85) {
		tmp = ((wj * wj) * (x + 1.0)) - (wj * x);
	} else if (wj <= 0.0039) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} else {
		tmp = wj + (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 <= (-1.45d-65)) then
        tmp = wj - ((wj - (x * (1.0d0 - wj))) / (wj + 1.0d0))
    else if (wj <= (-5.8d-85)) then
        tmp = ((wj * wj) * (x + 1.0d0)) - (wj * x)
    else if (wj <= 0.0039d0) then
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    else
        tmp = wj + (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 <= -1.45e-65) {
		tmp = wj - ((wj - (x * (1.0 - wj))) / (wj + 1.0));
	} else if (wj <= -5.8e-85) {
		tmp = ((wj * wj) * (x + 1.0)) - (wj * x);
	} else if (wj <= 0.0039) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} else {
		tmp = wj + (wj * (-1.0 / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.45e-65:
		tmp = wj - ((wj - (x * (1.0 - wj))) / (wj + 1.0))
	elif wj <= -5.8e-85:
		tmp = ((wj * wj) * (x + 1.0)) - (wj * x)
	elif wj <= 0.0039:
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	else:
		tmp = wj + (wj * (-1.0 / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.45e-65)
		tmp = Float64(wj - Float64(Float64(wj - Float64(x * Float64(1.0 - wj))) / Float64(wj + 1.0)));
	elseif (wj <= -5.8e-85)
		tmp = Float64(Float64(Float64(wj * wj) * Float64(x + 1.0)) - Float64(wj * x));
	elseif (wj <= 0.0039)
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	else
		tmp = Float64(wj + Float64(wj * Float64(-1.0 / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.45e-65)
		tmp = wj - ((wj - (x * (1.0 - wj))) / (wj + 1.0));
	elseif (wj <= -5.8e-85)
		tmp = ((wj * wj) * (x + 1.0)) - (wj * x);
	elseif (wj <= 0.0039)
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	else
		tmp = wj + (wj * (-1.0 / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.45e-65], N[(wj - N[(N[(wj - N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, -5.8e-85], N[(N[(N[(wj * wj), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(wj * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.0039], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if wj < -1.4499999999999999e-65

   1. Initial program 73.7%

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

      1. sub-neg 73.7%

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

      2. div-sub 73.7%

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

      3. sub-neg 73.7%

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

      4. +-commutative 73.7%

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

      5. distribute-neg-in 73.7%

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

      6. remove-double-neg 73.7%

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

      7. sub-neg 73.7%

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

      8. div-sub 73.7%

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

      9. distribute-rgt1-in 78.9%

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

      10. associate-/l/ 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in wj around 0 74.0%

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

      1. associate-*r* 74.0%

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

      2. neg-mul-1 74.0%

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

      3. distribute-lft1-in 73.9%

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

      4. +-commutative 73.9%

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

      5. sub-neg 73.9%

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

   6. Simplified 73.9%

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

   if -1.4499999999999999e-65 < wj < -5.8000000000000004e-85

   1. Initial program 18.3%

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

      1. sub-neg 18.3%

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

      2. div-sub 18.3%

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

      3. sub-neg 18.3%

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

      4. +-commutative 18.3%

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

      5. distribute-neg-in 18.3%

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

      6. remove-double-neg 18.3%

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

      7. sub-neg 18.3%

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

      8. div-sub 18.3%

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

      9. distribute-rgt1-in 18.3%

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

      10. associate-/l/ 18.3%

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

   3. Simplified 18.3%

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

      1. clear-num 18.0%

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

      2. associate-/r/ 18.3%

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

   5. Applied egg-rr 18.3%

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

   6. Taylor expanded in wj around 0 18.3%

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

      1. associate-*r* 18.3%

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

      2. neg-mul-1 18.3%

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

      3. distribute-lft1-in 18.3%

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

      4. +-commutative 18.3%

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

      5. sub-neg 18.3%

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

   8. Simplified 18.3%

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

   9. Taylor expanded in wj around inf 4.9%

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

       1. associate-*r* 4.9%

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

       2. mul-1-neg 4.9%

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

   11. Simplified 4.9%

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

   12. Taylor expanded in wj around 0 86.6%

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

       1. mul-1-neg 86.6%

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

       2. unsub-neg 86.6%

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

       3. +-commutative 86.6%

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

       4. unpow2 86.6%

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

       5. *-commutative 86.6%

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

   14. Simplified 86.6%

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

   if -5.8000000000000004e-85 < wj < 0.0038999999999999998

   1. Initial program 82.4%

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

      1. sub-neg 82.4%

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

      2. div-sub 82.4%

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

      3. sub-neg 82.4%

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

      4. +-commutative 82.4%

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

      5. distribute-neg-in 82.4%

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

      6. remove-double-neg 82.4%

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

      7. sub-neg 82.4%

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

      8. div-sub 82.4%

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

      9. distribute-rgt1-in 82.4%

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

      10. associate-/l/ 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in wj around 0 82.2%

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

      1. associate-*r* 82.2%

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

      2. neg-mul-1 82.2%

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

      3. distribute-lft1-in 82.2%

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

      4. +-commutative 82.2%

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

      5. sub-neg 82.2%

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

   6. Simplified 82.2%

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

   7. Taylor expanded in x around -inf 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-/l* 92.8%

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

      3. +-commutative 92.8%

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

   9. Simplified 92.8%

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

   if 0.0038999999999999998 < wj

   1. Initial program 63.8%

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

      1. sub-neg 63.8%

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

      2. div-sub 63.8%

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

      3. sub-neg 63.8%

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

      4. +-commutative 63.8%

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

      5. distribute-neg-in 63.8%

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

      6. remove-double-neg 63.8%

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

      7. sub-neg 63.8%

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

      8. div-sub 63.8%

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

      9. distribute-rgt1-in 63.8%

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

      10. associate-/l/ 63.9%

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

   3. Simplified 97.2%

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

      1. clear-num 97.2%

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

      2. associate-/r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in wj around 0 39.8%

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

      1. associate-*r* 39.5%

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

      2. neg-mul-1 39.5%

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

      3. distribute-lft1-in 39.5%

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

      4. +-commutative 39.5%

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

      5. sub-neg 39.5%

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

   8. Simplified 39.8%

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

   9. Taylor expanded in wj around inf 32.8%

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

       1. associate-*r* 32.8%

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

       2. mul-1-neg 32.8%

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

   11. Simplified 32.8%

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

   12. Taylor expanded in x around 0 81.9%

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

       1. mul-1-neg 81.9%

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

   14. Simplified 81.9%

       \[\leadsto wj + \frac{1}{wj + 1} \cdot \color{blue}{\left(-wj\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;wj - \frac{wj - x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{elif}\;wj \leq -5.8 \cdot 10^{-85}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(x + 1\right) - wj \cdot x\\ \mathbf{elif}\;wj \leq 0.0039:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj + wj \cdot \frac{-1}{wj + 1}\\ \end{array} \]

Alternative 6: 86.6% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0039:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0039) (/ x (/ (+ wj 1.0) (- 1.0 wj))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0039) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} 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 <= 0.0039d0) then
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    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 <= 0.0039) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.0039:
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0039)
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	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 <= 0.0039)
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0039], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0038999999999999998

   1. Initial program 80.0%

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

      1. sub-neg 80.0%

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

      2. div-sub 80.0%

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

      3. sub-neg 80.0%

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

      4. +-commutative 80.0%

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

      5. distribute-neg-in 80.0%

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

      6. remove-double-neg 80.0%

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

      7. sub-neg 80.0%

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

      8. div-sub 80.0%

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

      9. distribute-rgt1-in 80.4%

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

      10. associate-/l/ 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in wj around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-lft1-in 79.8%

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

      4. +-commutative 79.8%

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

      5. sub-neg 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in x around -inf 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-/l* 88.7%

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

      3. +-commutative 88.7%

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

   9. Simplified 88.7%

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

   if 0.0038999999999999998 < wj

   1. Initial program 63.8%

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

      1. sub-neg 63.8%

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

      2. div-sub 63.8%

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

      3. sub-neg 63.8%

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

      4. +-commutative 63.8%

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

      5. distribute-neg-in 63.8%

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

      6. remove-double-neg 63.8%

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

      7. sub-neg 63.8%

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

      8. div-sub 63.8%

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

      9. distribute-rgt1-in 63.8%

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

      10. associate-/l/ 63.9%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in x around 0 81.6%

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

      1. +-commutative 81.6%

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

   6. Simplified 81.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0039:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 7: 86.6% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0039:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj + wj \cdot \frac{-1}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0039)
   (/ x (/ (+ wj 1.0) (- 1.0 wj)))
   (+ wj (* wj (/ -1.0 (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0039) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} else {
		tmp = wj + (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 <= 0.0039d0) then
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    else
        tmp = wj + (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 <= 0.0039) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} else {
		tmp = wj + (wj * (-1.0 / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.0039:
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	else:
		tmp = wj + (wj * (-1.0 / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0039)
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	else
		tmp = Float64(wj + Float64(wj * Float64(-1.0 / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.0039)
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	else
		tmp = wj + (wj * (-1.0 / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0039], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0038999999999999998

   1. Initial program 80.0%

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

      1. sub-neg 80.0%

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

      2. div-sub 80.0%

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

      3. sub-neg 80.0%

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

      4. +-commutative 80.0%

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

      5. distribute-neg-in 80.0%

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

      6. remove-double-neg 80.0%

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

      7. sub-neg 80.0%

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

      8. div-sub 80.0%

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

      9. distribute-rgt1-in 80.4%

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

      10. associate-/l/ 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in wj around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-lft1-in 79.8%

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

      4. +-commutative 79.8%

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

      5. sub-neg 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in x around -inf 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-/l* 88.7%

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

      3. +-commutative 88.7%

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

   9. Simplified 88.7%

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

   if 0.0038999999999999998 < wj

   1. Initial program 63.8%

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

      1. sub-neg 63.8%

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

      2. div-sub 63.8%

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

      3. sub-neg 63.8%

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

      4. +-commutative 63.8%

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

      5. distribute-neg-in 63.8%

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

      6. remove-double-neg 63.8%

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

      7. sub-neg 63.8%

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

      8. div-sub 63.8%

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

      9. distribute-rgt1-in 63.8%

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

      10. associate-/l/ 63.9%

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

   3. Simplified 97.2%

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

      1. clear-num 97.2%

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

      2. associate-/r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in wj around 0 39.8%

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

      1. associate-*r* 39.5%

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

      2. neg-mul-1 39.5%

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

      3. distribute-lft1-in 39.5%

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

      4. +-commutative 39.5%

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

      5. sub-neg 39.5%

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

   8. Simplified 39.8%

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

   9. Taylor expanded in wj around inf 32.8%

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

       1. associate-*r* 32.8%

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

       2. mul-1-neg 32.8%

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

   11. Simplified 32.8%

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

   12. Taylor expanded in x around 0 81.9%

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

       1. mul-1-neg 81.9%

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

   14. Simplified 81.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0039:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj + wj \cdot \frac{-1}{wj + 1}\\ \end{array} \]

Alternative 8: 95.7% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. sub-neg 79.6%

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

   2. div-sub 79.6%

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

   3. sub-neg 79.6%

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

   4. +-commutative 79.6%

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

   5. distribute-neg-in 79.6%

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

   6. remove-double-neg 79.6%

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

   7. sub-neg 79.6%

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

   8. div-sub 79.6%

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

   9. distribute-rgt1-in 80.0%

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

   10. associate-/l/ 80.0%

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

3. Simplified 80.7%

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

4. Taylor expanded in wj around 0 78.8%

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

   1. associate-*r* 78.8%

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

   2. neg-mul-1 78.8%

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

   3. distribute-lft1-in 78.8%

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

   4. +-commutative 78.8%

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

   5. sub-neg 78.8%

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

6. Simplified 78.8%

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

7. Taylor expanded in wj around 0 97.0%

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

8. Taylor expanded in x around 0 97.0%

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

   1. unpow2 97.0%

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

10. Simplified 97.0%

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

11. Final simplification 97.0%

    \[\leadsto wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right) \]

Alternative 9: 86.6% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0039:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0039) (+ x (* -2.0 (* wj x))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0039) {
		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 <= 0.0039d0) 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 <= 0.0039) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.0039:
		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 <= 0.0039)
		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 <= 0.0039)
		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, 0.0039], 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 < 0.0038999999999999998

   1. Initial program 80.0%

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

      1. sub-neg 80.0%

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

      2. div-sub 80.0%

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

      3. sub-neg 80.0%

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

      4. +-commutative 80.0%

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

      5. distribute-neg-in 80.0%

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

      6. remove-double-neg 80.0%

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

      7. sub-neg 80.0%

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

      8. div-sub 80.0%

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

      9. distribute-rgt1-in 80.4%

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

      10. associate-/l/ 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in wj around 0 88.7%

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

   if 0.0038999999999999998 < wj

   1. Initial program 63.8%

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

      1. sub-neg 63.8%

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

      2. div-sub 63.8%

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

      3. sub-neg 63.8%

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

      4. +-commutative 63.8%

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

      5. distribute-neg-in 63.8%

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

      6. remove-double-neg 63.8%

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

      7. sub-neg 63.8%

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

      8. div-sub 63.8%

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

      9. distribute-rgt1-in 63.8%

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

      10. associate-/l/ 63.9%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in x around 0 81.6%

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

      1. +-commutative 81.6%

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

   6. Simplified 81.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0039:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 10: 85.3% 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 79.6%

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

   1. sub-neg 79.6%

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

   2. div-sub 79.6%

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

   3. sub-neg 79.6%

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

   4. +-commutative 79.6%

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

   5. distribute-neg-in 79.6%

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

   6. remove-double-neg 79.6%

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

   7. sub-neg 79.6%

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

   8. div-sub 79.6%

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

   9. distribute-rgt1-in 80.0%

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

   10. associate-/l/ 80.0%

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

3. Simplified 80.7%

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

4. Taylor expanded in wj around 0 86.8%

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

5. Final simplification 86.8%

   \[\leadsto x + -2 \cdot \left(wj \cdot x\right) \]

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 79.6%

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

   1. sub-neg 79.6%

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

   2. div-sub 79.6%

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

   3. sub-neg 79.6%

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

   4. +-commutative 79.6%

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

   5. distribute-neg-in 79.6%

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

   6. remove-double-neg 79.6%

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

   7. sub-neg 79.6%

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

   8. div-sub 79.6%

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

   9. distribute-rgt1-in 80.0%

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

   10. associate-/l/ 80.0%

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

3. Simplified 80.7%

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

4. Taylor expanded in wj around inf 4.4%

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

5. Final simplification 4.4%

   \[\leadsto wj \]

Alternative 12: 84.7% 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 79.6%

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

   1. sub-neg 79.6%

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

   2. div-sub 79.6%

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

   3. sub-neg 79.6%

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

   4. +-commutative 79.6%

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

   5. distribute-neg-in 79.6%

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

   6. remove-double-neg 79.6%

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

   7. sub-neg 79.6%

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

   8. div-sub 79.6%

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

   9. distribute-rgt1-in 80.0%

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

   10. associate-/l/ 80.0%

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

3. Simplified 80.7%

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

4. Taylor expanded in wj around 0 86.4%

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

5. Final simplification 86.4%

   \[\leadsto x \]

Developer target: 79.4% 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 2023193 
(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))))))