Average Error: 13.7 → 1.0
Time: 4.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 8.74113048810989798 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x - 2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{\frac{x}{wj \cdot wj - 1}}{\frac{e^{wj}}{wj - 1}} - \frac{wj}{wj + 1}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 8.74113048810989798 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(wj, wj, x - 2 \cdot \left(wj \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;wj + \left(\frac{\frac{x}{wj \cdot wj - 1}}{\frac{e^{wj}}{wj - 1}} - \frac{wj}{wj + 1}\right)\\

\end{array}
double code(double wj, double x) {
	return (wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj)))));
}
↓
double code(double wj, double x) {
	double VAR;
	if ((wj <= 8.741130488109898e-09)) {
		VAR = fma(wj, wj, (x - (2.0 * (wj * x))));
	} else {
		VAR = (wj + (((x / ((wj * wj) - 1.0)) / (exp(wj) / (wj - 1.0))) - (wj / (wj + 1.0))));
	}
	return VAR;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.2
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 8.741130488109898e-09

    1. Initial program 13.4

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

    2. Simplified13.4

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified0.9

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

    if 8.741130488109898e-09 < wj

    1. Initial program 23.9

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

    2. Simplified3.8

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

    4. Applied +-commutative3.8

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

    5. Applied associate--l+3.8

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

    7. Applied flip-+3.8

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

    8. Applied associate-/r/3.8

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

    9. Applied associate-/l*3.8

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.74113048810989798 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x - 2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{\frac{x}{wj \cdot wj - 1}}{\frac{e^{wj}}{wj - 1}} - \frac{wj}{wj + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))