Average Error: 13.9 → 0.9
Time: 4.7s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2.233670756457821 \cdot 10^{-18}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\sqrt{wj + 1}} \cdot \frac{\frac{x}{e^{wj}} - wj}{\sqrt{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2.233670756457821 \cdot 10^{-18}:\\
\;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\

\mathbf{else}:\\
\;\;\;\;wj + \frac{1}{\sqrt{wj + 1}} \cdot \frac{\frac{x}{e^{wj}} - wj}{\sqrt{wj + 1}}\\

\end{array}
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (if (<=
      (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
      2.233670756457821e-18)
   (+ x (* wj (+ wj (* x -2.0))))
   (+
    wj
    (*
     (/ 1.0 (sqrt (+ wj 1.0)))
     (/ (- (/ x (exp wj)) wj) (sqrt (+ wj 1.0)))))))
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 - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))))) <= 2.233670756457821e-18) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + ((1.0 / sqrt(wj + 1.0)) * (((x / exp(wj)) - wj) / sqrt(wj + 1.0)));
	}
	return tmp;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
Target13.2
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

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))))) < 2.23367076e-18

    1. Initial program 18.3

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

    2. Simplified18.3

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

    3. Taylor expanded around 0 0.8

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

    4. Simplified0.8

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

    if 2.23367076e-18 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

    1. Initial program 3.1

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

    2. Simplified0.7

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt_binary64_31691.0

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

    5. Applied *-un-lft-identity_binary64_31471.0

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

    6. Applied times-frac_binary64_31531.0

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2.233670756457821 \cdot 10^{-18}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\sqrt{wj + 1}} \cdot \frac{\frac{x}{e^{wj}} - wj}{\sqrt{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(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))))))