Jmat.Real.lambertw, newton loop step

Percentage Accurate: 39.7% → 99.3%

Time: 12.2s

Precision: binary64

Cost: 15816

• Unsound rule application detected in e-graph. Results may not be sound.

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -0.00013:\\ \;\;\;\;\frac{e^{-wj} \cdot x}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.049:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot t_0\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -0.00013)
     (/ (* (exp (- wj)) x) (+ wj 1.0))
     (if (<= wj 0.049)
       (+
        (*
         (pow wj 3.0)
         (- (- (- -1.0 (* -2.0 t_0)) (* x -3.0)) (* x 0.6666666666666666)))
        (+ (* (- 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* wj x)))))
       (- wj (/ wj (+ wj 1.0)))))))

C

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

↓

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -0.00013) {
		tmp = (exp(-wj) * x) / (wj + 1.0);
	} else if (wj <= 0.049) {
		tmp = (pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * pow(wj, 2.0)) + (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
    code = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))))
end function

↓

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 * (-4.0d0)) + (x * 1.5d0)
    if (wj <= (-0.00013d0)) then
        tmp = (exp(-wj) * x) / (wj + 1.0d0)
    else if (wj <= 0.049d0) then
        tmp = ((wj ** 3.0d0) * ((((-1.0d0) - ((-2.0d0) * t_0)) - (x * (-3.0d0))) - (x * 0.6666666666666666d0))) + (((1.0d0 - t_0) * (wj ** 2.0d0)) + (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) {
	return wj - (((wj * Math.exp(wj)) - x) / (Math.exp(wj) + (wj * Math.exp(wj))));
}

↓

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -0.00013) {
		tmp = (Math.exp(-wj) * x) / (wj + 1.0);
	} else if (wj <= 0.049) {
		tmp = (Math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * Math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

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

↓

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= -0.00013:
		tmp = (math.exp(-wj) * x) / (wj + 1.0)
	elif wj <= 0.049:
		tmp = (math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

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

↓

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= -0.00013)
		tmp = Float64(Float64(exp(Float64(-wj)) * x) / Float64(wj + 1.0));
	elseif (wj <= 0.049)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * t_0)) - Float64(x * -3.0)) - Float64(x * 0.6666666666666666))) + Float64(Float64(Float64(1.0 - t_0) * (wj ^ 2.0)) + 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 = code(wj, x)
	tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
end

↓

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= -0.00013)
		tmp = (exp(-wj) * x) / (wj + 1.0);
	elseif (wj <= 0.049)
		tmp = ((wj ^ 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * (wj ^ 2.0)) + (x + (-2.0 * (wj * x))));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -0.00013], N[(N[(N[Exp[(-wj)], $MachinePrecision] * x), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.049], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	39.7%
Target	64.2%
Herbie	99.3%

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

Derivation

1. Split input into 3 regimes

2. if wj < -1.29999999999999989e-4

   1. Initial program 2.6%

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

   2. Simplified 100.0%

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

      [Start] 2.6%	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 2.6%	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 2.6%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [=>] 2.6%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} + \left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]
      +-commutative [=>] 2.6%	\[ wj + \left(-\color{blue}{\left(\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      distribute-neg-in [=>] 2.6%	\[ wj + \color{blue}{\left(\left(-\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right) + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      remove-double-neg [=>] 2.6%	\[ wj + \left(\color{blue}{\frac{x}{e^{wj} + wj \cdot e^{wj}}} + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right) \]
      sub-neg [<=] 2.6%	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [<=] 2.6%	\[ wj + \color{blue}{\frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
      distribute-rgt1-in [=>] 100.0%	\[ wj + \frac{x - wj \cdot e^{wj}}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]
      associate-/l/ [<=] 100.0%	\[ wj + \color{blue}{\frac{\frac{x - wj \cdot e^{wj}}{e^{wj}}}{wj + 1}} \]

   3. Applied egg-rr 100.0%

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

      [Start] 100.0%	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
      clear-num [=>] 100.0%	\[ wj + \frac{\color{blue}{\frac{1}{\frac{e^{wj}}{x}}} - wj}{wj + 1} \]
      associate-/r/ [=>] 100.0%	\[ wj + \frac{\color{blue}{\frac{1}{e^{wj}} \cdot x} - wj}{wj + 1} \]
      rec-exp [=>] 100.0%	\[ wj + \frac{\color{blue}{e^{-wj}} \cdot x - wj}{wj + 1} \]

   4. Taylor expanded in x around inf 100.0%

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

   if -1.29999999999999989e-4 < wj < 0.049000000000000002

   1. Initial program 79.0%

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

   2. Simplified 79.0%

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

      [Start] 79.0%	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 79.0%	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 79.0%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [=>] 79.0%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} + \left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]
      +-commutative [=>] 79.0%	\[ wj + \left(-\color{blue}{\left(\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      distribute-neg-in [=>] 79.0%	\[ wj + \color{blue}{\left(\left(-\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right) + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      remove-double-neg [=>] 79.0%	\[ wj + \left(\color{blue}{\frac{x}{e^{wj} + wj \cdot e^{wj}}} + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right) \]
      sub-neg [<=] 79.0%	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [<=] 79.0%	\[ wj + \color{blue}{\frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
      distribute-rgt1-in [=>] 79.0%	\[ wj + \frac{x - wj \cdot e^{wj}}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]
      associate-/l/ [<=] 79.0%	\[ wj + \color{blue}{\frac{\frac{x - wj \cdot e^{wj}}{e^{wj}}}{wj + 1}} \]

   3. Taylor expanded in wj around 0 99.9%

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

   if 0.049000000000000002 < wj

   1. Initial program 0.0%

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

   2. Simplified 100.0%

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

      [Start] 0.0%	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 0.0%	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 0.0%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [=>] 0.0%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} + \left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]
      +-commutative [=>] 0.0%	\[ wj + \left(-\color{blue}{\left(\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      distribute-neg-in [=>] 0.0%	\[ wj + \color{blue}{\left(\left(-\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right) + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      remove-double-neg [=>] 0.0%	\[ wj + \left(\color{blue}{\frac{x}{e^{wj} + wj \cdot e^{wj}}} + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right) \]
      sub-neg [<=] 0.0%	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [<=] 0.0%	\[ wj + \color{blue}{\frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
      distribute-rgt1-in [=>] 0.0%	\[ wj + \frac{x - wj \cdot e^{wj}}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]
      associate-/l/ [<=] 0.0%	\[ wj + \color{blue}{\frac{\frac{x - wj \cdot e^{wj}}{e^{wj}}}{wj + 1}} \]

   3. Taylor expanded in x around 0 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0%	\[ wj - \frac{wj}{1 + wj} \]
      +-commutative [<=] 100.0%	\[ wj - \frac{wj}{\color{blue}{wj + 1}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.00013:\\ \;\;\;\;\frac{e^{-wj} \cdot x}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.049:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	15816

\[\begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -0.00013:\\ \;\;\;\;\frac{e^{-wj} \cdot x}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.049:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot t_0\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;wj \leq -9.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{-wj} \cdot x}{wj + 1}\\ \mathbf{elif}\;wj \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;wj \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{-wj} \cdot x}{wj + 1}\\ \mathbf{elif}\;wj \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 4
Accuracy	98.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;wj \leq -9.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 5
Accuracy	98.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;wj \leq -1:\\ \;\;\;\;\frac{x}{wj \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 6
Accuracy	82.7%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;wj \leq -1:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(-wj\right)\right)\\ \mathbf{elif}\;wj \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 7
Accuracy	82.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;wj \leq -1:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(-wj\right)\right)\\ \mathbf{elif}\;wj \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8
Accuracy	76.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;wj \leq -0.6:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(-wj\right)\right)\\ \mathbf{elif}\;wj \leq 0.91:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \]

Alternative 9
Accuracy	77.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;wj \leq -0.41:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(-wj\right)\right)\\ \mathbf{elif}\;wj \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 10
Accuracy	77.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;wj \leq -2.25 \cdot 10^{+49}:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(-wj\right)\right)\\ \mathbf{elif}\;wj \leq 2.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 11
Accuracy	74.4%
Cost	708

\[\begin{array}{l} \mathbf{if}\;wj \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 12
Accuracy	76.5%
Cost	516

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.15:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(-wj\right)\right)\\ \mathbf{elif}\;wj \leq 3.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \]

Alternative 13
Accuracy	74.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;wj \leq -9.5 \cdot 10^{-17}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 1.85:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \]

Alternative 14
Accuracy	73.9%
Cost	328

\[\begin{array}{l} \mathbf{if}\;wj \leq -6.5 \cdot 10^{-18}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj\\ \end{array} \]

Alternative 15
Accuracy	67.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;wj \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj\\ \end{array} \]

Alternative 16
Accuracy	2.4%
Cost	64

\[-512 \]

Alternative 17
Accuracy	2.8%
Cost	64

\[0 \]

Alternative 18
Accuracy	3.8%
Cost	64

\[0.5 \]

Alternative 19
Accuracy	3.8%
Cost	64

\[19683 \]

Alternative 20
Accuracy	26.9%
Cost	64

\[wj \]

Reproduce

herbie shell --seed 2023272 
(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))))))