Average Error: 14.0 → 0.4
Time: 4.8s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq -4.453837834497543 \lor \neg \left(wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2.7132881184931778 \cdot 10^{-21}\right):\\ \;\;\;\;\left(wj + \frac{\frac{x}{e^{wj}}}{wj + 1}\right) - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \end{array}\]
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 -4.453837834497543 \lor \neg \left(wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2.7132881184931778 \cdot 10^{-21}\right):\\
\;\;\;\;\left(wj + \frac{\frac{x}{e^{wj}}}{wj + 1}\right) - \frac{wj}{wj + 1}\\

\mathbf{else}:\\
\;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\

\end{array}
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (if (or (<=
          (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
          -4.453837834497543)
         (not
          (<=
           (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
           2.7132881184931778e-21)))
   (- (+ wj (/ (/ x (exp wj)) (+ wj 1.0))) (/ wj (+ wj 1.0)))
   (+ x (* wj (+ wj (* x -2.0))))))
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))))) <= -4.453837834497543) || !((wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))))) <= 2.7132881184931778e-21)) {
		tmp = (wj + ((x / exp(wj)) / (wj + 1.0))) - (wj / (wj + 1.0));
	} else {
		tmp = x + (wj * (wj + (x * -2.0)));
	}
	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

Original14.0
Target13.3
Herbie0.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

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))))) < -4.4538378344975431 or 2.7132881e-21 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

    1. Initial program 1.9

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

    2. Simplified0.6

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

    4. Applied div-sub_binary64_38340.6

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

    5. Applied associate-+r-_binary64_37630.6

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

    if -4.4538378344975431 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2.7132881e-21

    1. Initial program 28.0

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

    2. Simplified27.9

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

    3. Taylor expanded around 0 0.2

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

    4. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq -4.453837834497543 \lor \neg \left(wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2.7132881184931778 \cdot 10^{-21}\right):\\ \;\;\;\;\left(wj + \frac{\frac{x}{e^{wj}}}{wj + 1}\right) - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(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))))))