Average Error: 13.9 → 1.1
Time: 3.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)
double code(double wj, double x) {
	return (wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj)))));
}
↓
double code(double wj, double x) {
	return ((x / (exp(wj) * (wj + 1.0))) + ((pow(wj, 4.0) + pow(wj, 2.0)) - pow(wj, 3.0)));
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
Target13.3
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.9

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

  2. Simplified13.3

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

  4. Applied associate--l+7.3

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

  5. Taylor expanded around 0 1.1

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

  7. Applied div-inv1.1

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

  8. Applied associate-/l*1.1

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

  9. Simplified1.1

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

  10. Final simplification1.1

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

Reproduce

herbie shell --seed 2020071 
(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))))))