Average Error: 13.4 → 0.6
Time: 6.9s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq 9.953616011917723 \cdot 10^{-07}:\\ \;\;\;\;x + \left(\left(wj \cdot wj\right) \cdot \left(\left(x \cdot 2.5 + 1\right) - wj\right) - x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq 9.953616011917723 \cdot 10^{-07}:\\
\;\;\;\;x + \left(\left(wj \cdot wj\right) \cdot \left(\left(x \cdot 2.5 + 1\right) - wj\right) - x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)\\

\mathbf{else}:\\
\;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{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 9.953616011917723e-07)
   (+
    x
    (-
     (* (* wj wj) (- (+ (* x 2.5) 1.0) wj))
     (* x (+ (+ wj wj) (* (pow wj 3.0) 2.6666666666666665)))))
   (+ wj (/ (- (* x (exp (- wj))) wj) (+ 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 <= 9.953616011917723e-07) {
		tmp = x + (((wj * wj) * (((x * 2.5) + 1.0) - wj)) - (x * ((wj + wj) + (pow(wj, 3.0) * 2.6666666666666665))));
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (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

Original13.4
Target12.8
Herbie0.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 9.953616011917723e-7

    1. Initial program 13.1

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

    2. Simplified13.1

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

    3. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{\left({wj}^{2} + \left(x + 2.5 \cdot \left({wj}^{2} \cdot x\right)\right)\right) - \left({wj}^{3} + \left(2.6666666666666665 \cdot \left({wj}^{3} \cdot x\right) + 2 \cdot \left(wj \cdot x\right)\right)\right)}\]

    4. Simplified0.6

      \[\leadsto \color{blue}{x + \left(\left(2.5 \cdot x + 1\right) \cdot \left(wj \cdot wj\right) - \left({wj}^{3} + x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)\right)}\]
    5. Using strategy rm

    6. Applied associate--r+_binary64_37650.6

      \[\leadsto x + \color{blue}{\left(\left(\left(2.5 \cdot x + 1\right) \cdot \left(wj \cdot wj\right) - {wj}^{3}\right) - x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)}\]

    7. Simplified0.6

      \[\leadsto x + \left(\color{blue}{\left(wj \cdot wj\right) \cdot \left(\left(2.5 \cdot x + 1\right) - wj\right)} - x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)\]

    if 9.953616011917723e-7 < wj

    1. Initial program 28.2

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

    2. Simplified1.7

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

    4. Applied div-inv_binary64_38261.7

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

    5. Simplified1.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 9.953616011917723 \cdot 10^{-07}:\\ \;\;\;\;x + \left(\left(wj \cdot wj\right) \cdot \left(\left(x \cdot 2.5 + 1\right) - wj\right) - x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020357 
(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))))))