Average Error: 14.3 → 1.5
Time: 4.0s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -7.962081679886866 \cdot 10^{-17}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -7.962081679886866 \cdot 10^{-17}:\\
\;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\

\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 (<= x -7.962081679886866e-17)
   (+ wj (* (/ (- (/ x (exp wj)) wj) (- (* wj wj) 1.0)) (- wj 1.0)))
   (+ x (* wj (+ wj (* x -2.0))))))
double code(double wj, double x) {
	return ((double) (wj - (((double) (((double) (wj * ((double) exp(wj)))) - x)) / ((double) (((double) exp(wj)) + ((double) (wj * ((double) exp(wj)))))))));
}
↓
double code(double wj, double x) {
	double tmp;
	if ((x <= -7.962081679886866e-17)) {
		tmp = ((double) (wj + ((double) ((((double) ((x / ((double) exp(wj))) - wj)) / ((double) (((double) (wj * wj)) - 1.0))) * ((double) (wj - 1.0))))));
	} else {
		tmp = ((double) (x + ((double) (wj * ((double) (wj + ((double) (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

Original14.3
Target13.8
Herbie1.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.9620816798868662e-17

    1. Initial program 0.6

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

    2. Simplified0.1

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

    4. Applied flip-+_binary640.1

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

    5. Applied associate-/r/_binary640.1

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

    6. Simplified0.1

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

    if -7.9620816798868662e-17 < x

    1. Initial program 19.2

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

    2. Simplified18.7

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

    3. Taylor expanded around 0 2.0

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

    4. Simplified2.0

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.962081679886866 \cdot 10^{-17}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))))))