Average Error: 13.5 → 1.2
Time: 6.4s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{wj + 1}}{e^{wj}} + \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}} \cdot \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{wj + 1}}{e^{wj}} + \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}} \cdot \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}}
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) (x / ((double) (wj + 1.0)))) / ((double) exp(wj)))) + ((double) (((double) sqrt(((double) (((double) (((double) pow(wj, 4.0)) + ((double) pow(wj, 2.0)))) - ((double) pow(wj, 3.0)))))) * ((double) sqrt(((double) (((double) (((double) pow(wj, 4.0)) + ((double) pow(wj, 2.0)))) - ((double) pow(wj, 3.0))))))))));
}

Error

Bits error versus wj

Bits error versus x

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Results

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Target

Original13.5
Target12.8
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

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

  2. Simplified12.8

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

  4. Applied associate--l+6.8

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

  5. Taylor expanded around 0 1.2

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

  7. Applied add-sqr-sqrt1.2

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

  8. Final simplification1.2

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

Reproduce

herbie shell --seed 2020162 
(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))))))