Jmat.Real.lambertw, newton loop step

Average Error: 14.2 → 0.6

Time: 17.8s

Precision: binary64

Cost: 19972

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2 - wj\right) \cdot \left(\left(\frac{x}{e^{wj}} - wj\right) + wj \cdot \left(wj + 1\right)\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.7e-7)
   (- (fma wj wj (fma -2.0 (* wj x) x)) (pow wj 3.0))
   (/
    (* (- -2.0 wj) (+ (- (/ x (exp wj)) wj) (* wj (+ wj 1.0))))
    (* (+ wj 2.0) (- -1.0 wj)))))

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 tmp;
	if (wj <= 3.7e-7) {
		tmp = fma(wj, wj, fma(-2.0, (wj * x), x)) - pow(wj, 3.0);
	} else {
		tmp = ((-2.0 - wj) * (((x / exp(wj)) - wj) + (wj * (wj + 1.0)))) / ((wj + 2.0) * (-1.0 - wj));
	}
	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)
	tmp = 0.0
	if (wj <= 3.7e-7)
		tmp = Float64(fma(wj, wj, fma(-2.0, Float64(wj * x), x)) - (wj ^ 3.0));
	else
		tmp = Float64(Float64(Float64(-2.0 - wj) * Float64(Float64(Float64(x / exp(wj)) - wj) + Float64(wj * Float64(wj + 1.0)))) / Float64(Float64(wj + 2.0) * Float64(-1.0 - wj)));
	end
	return 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_] := If[LessEqual[wj, 3.7e-7], N[(N[(wj * wj + N[(-2.0 * N[(wj * x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 - wj), $MachinePrecision] * N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] + N[(wj * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(wj + 2.0), $MachinePrecision] * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	14.2
Target	13.5
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if wj < 3.70000000000000004e-7

   1. Initial program 13.8

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

   2. Simplified 13.8

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

      [Start] 13.8	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 13.8	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 13.8	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 13.8	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 13.8	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 13.8	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 13.8	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 13.8	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 13.8	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 13.8	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 13.8	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 13.8	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Taylor expanded in wj around 0 0.6

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

   4. Taylor expanded in x around 0 0.7

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

   5. Simplified 0.7

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

      [Start] 0.7	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left({wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      unpow2 [=>] 0.7	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\color{blue}{wj \cdot wj} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]

   6. Taylor expanded in x around 0 0.6

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

   7. Taylor expanded in wj around 0 0.6

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

   8. Simplified 0.6

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

      [Start] 0.6	\[ -1 \cdot {wj}^{3} + \left({wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      unpow2 [=>] 0.6	\[ -1 \cdot {wj}^{3} + \left(\color{blue}{wj \cdot wj} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      fma-def [=>] 0.6	\[ -1 \cdot {wj}^{3} + \left(wj \cdot wj + \color{blue}{\mathsf{fma}\left(-2, wj \cdot x, x\right)}\right) \]
      fma-udef [<=] 0.6	\[ -1 \cdot {wj}^{3} + \color{blue}{\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)} \]
      *-commutative [=>] 0.6	\[ -1 \cdot {wj}^{3} + \mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, \color{blue}{x \cdot wj}, x\right)\right) \]

   if 3.70000000000000004e-7 < wj

   1. Initial program 32.9

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

   2. Simplified 2.1

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

      [Start] 32.9	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 32.9	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 32.9	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 32.9	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 32.9	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 32.9	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 32.9	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 32.9	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 32.9	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 32.9	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 32.9	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 32.9	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Applied egg-rr 4.4

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

   4. Simplified 4.4

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

      [Start] 4.4	\[ \left(wj + 1\right) - \left(1 - \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right) \]
      associate--r- [=>] 4.4	\[ \color{blue}{\left(\left(wj + 1\right) - 1\right) + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]

   5. Applied egg-rr 2.4

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

   6. Simplified 2.4

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

      [Start] 2.4	\[ \frac{\left(\left(wj + 2\right) \cdot wj\right) \cdot \left(-1 - wj\right) + \left(wj + 2\right) \cdot \left(-\left(\frac{x}{e^{wj}} - wj\right)\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)} \]
      +-commutative [=>] 2.4	\[ \frac{\color{blue}{\left(wj + 2\right) \cdot \left(-\left(\frac{x}{e^{wj}} - wj\right)\right) + \left(\left(wj + 2\right) \cdot wj\right) \cdot \left(-1 - wj\right)}}{\left(wj + 2\right) \cdot \left(-1 - wj\right)} \]
      associate-*l* [=>] 2.4	\[ \frac{\left(wj + 2\right) \cdot \left(-\left(\frac{x}{e^{wj}} - wj\right)\right) + \color{blue}{\left(wj + 2\right) \cdot \left(wj \cdot \left(-1 - wj\right)\right)}}{\left(wj + 2\right) \cdot \left(-1 - wj\right)} \]
      distribute-lft-out [=>] 2.4	\[ \frac{\color{blue}{\left(wj + 2\right) \cdot \left(\left(-\left(\frac{x}{e^{wj}} - wj\right)\right) + wj \cdot \left(-1 - wj\right)\right)}}{\left(wj + 2\right) \cdot \left(-1 - wj\right)} \]
      neg-sub0 [=>] 2.4	\[ \frac{\left(wj + 2\right) \cdot \left(\color{blue}{\left(0 - \left(\frac{x}{e^{wj}} - wj\right)\right)} + wj \cdot \left(-1 - wj\right)\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)} \]
      associate--r- [=>] 2.4	\[ \frac{\left(wj + 2\right) \cdot \left(\color{blue}{\left(\left(0 - \frac{x}{e^{wj}}\right) + wj\right)} + wj \cdot \left(-1 - wj\right)\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)} \]
      neg-sub0 [<=] 2.4	\[ \frac{\left(wj + 2\right) \cdot \left(\left(\color{blue}{\left(-\frac{x}{e^{wj}}\right)} + wj\right) + wj \cdot \left(-1 - wj\right)\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)} \]
      distribute-neg-frac [=>] 2.4	\[ \frac{\left(wj + 2\right) \cdot \left(\left(\color{blue}{\frac{-x}{e^{wj}}} + wj\right) + wj \cdot \left(-1 - wj\right)\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2 - wj\right) \cdot \left(\left(\frac{x}{e^{wj}} - wj\right) + wj \cdot \left(wj + 1\right)\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	8004

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

Alternative 2
Error	0.6
Cost	7428

\[\begin{array}{l} \mathbf{if}\;wj \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;\left(wj \cdot wj + \left(x - 2 \cdot \left(wj \cdot x\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 3
Error	1.5
Cost	7300

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+36}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + wj \cdot wj\right) - {wj}^{3}\\ \end{array} \]

Alternative 4
Error	1.5
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+46}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + wj \cdot wj\right) - {wj}^{3}\\ \end{array} \]

Alternative 5
Error	8.6
Cost	7044

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.55 \cdot 10^{-38}:\\ \;\;\;\;\frac{x \cdot e^{-wj}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(wj \cdot wj\right) \cdot \left(-2 - wj\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)}\\ \end{array} \]

Alternative 6
Error	8.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;wj \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(wj \cdot wj\right) \cdot \left(-2 - wj\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)}\\ \end{array} \]

Alternative 7
Error	8.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.12 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(wj \cdot wj\right) \cdot \left(-2 - wj\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)}\\ \end{array} \]

Alternative 8
Error	2.0
Cost	6912

\[\left(x + wj \cdot wj\right) - {wj}^{3} \]

Alternative 9
Error	8.3
Cost	1220

\[\begin{array}{l} \mathbf{if}\;wj \leq 5.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(wj \cdot wj\right) \cdot \left(-2 - wj\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)}\\ \end{array} \]

Alternative 10
Error	8.8
Cost	1092

\[\begin{array}{l} \mathbf{if}\;wj \leq 4.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{1 - wj \cdot \left(-2 - wj \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot \left(wj \cdot \left(-2 - wj\right)\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)}\\ \end{array} \]

Alternative 11
Error	8.8
Cost	1092

\[\begin{array}{l} \mathbf{if}\;wj \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{1 - wj \cdot \left(-2 - wj \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(wj \cdot wj\right) \cdot \left(-2 - wj\right)}{\left(wj + 2\right) \cdot \left(-1 - wj\right)}\\ \end{array} \]

Alternative 12
Error	8.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{1 - wj \cdot \left(-2 - wj \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj \cdot \left(wj + -1\right)}{1 - wj \cdot wj}\\ \end{array} \]

Alternative 13
Error	8.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{1 - wj \cdot \left(-2 - wj \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 14
Error	8.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{1 - wj \cdot \left(-2 - wj \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot \left(wj + 1\right) - wj}{wj + 1}\\ \end{array} \]

Alternative 15
Error	8.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;x - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 16
Error	9.3
Cost	448

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

Alternative 17
Error	61.2
Cost	64

\[wj \]

Alternative 18
Error	9.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023066 
(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))))))