Average Error: 13.2 → 1.0
Time: 3.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}
double code(double wj, double x) {
	return ((double) (wj - ((double) (((double) (((double) (wj * ((double) exp(wj)))) - x)) / ((double) (((double) exp(wj)) + ((double) (wj * ((double) exp(wj))))))))));
}
↓
double code(double wj, double x) {
	return ((double) (((double) (((double) (((double) pow(wj, 4.0)) + ((double) pow(wj, 2.0)))) - ((double) pow(wj, 3.0)))) + ((double) (((double) (x / ((double) exp(wj)))) / ((double) (1.0 + wj))))));
}

Error

Bits error versus wj

Bits error versus x

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Results

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Target

Original13.2
Target12.7
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.2

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
  2. Using strategy rm

  3. Applied div-sub13.2

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

  4. Applied associate--r-7.2

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

  5. Simplified7.2

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

  6. Taylor expanded around 0 1.0

    \[\leadsto \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity1.0

    \[\leadsto \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{x}{\color{blue}{1 \cdot e^{wj}} + wj \cdot e^{wj}}\]

  9. Applied distribute-rgt-out1.0

    \[\leadsto \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{x}{\color{blue}{e^{wj} \cdot \left(1 + wj\right)}}\]

  10. Applied associate-/r*1.0

    \[\leadsto \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \color{blue}{\frac{\frac{x}{e^{wj}}}{1 + wj}}\]

  11. Final simplification1.0

    \[\leadsto \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]

Reproduce

herbie shell --seed 2020140 
(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))))))