Average Error: 13.5 → 1.2
Time: 11.5s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\frac{x}{e^{wj} \cdot \left(1 + wj\right)} + \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) \]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{x}{e^{wj} \cdot \left(1 + wj\right)} + \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right)

(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (+ (/ x (* (exp wj) (+ 1.0 wj))) (- (fma wj wj (pow wj 4.0)) (pow wj 3.0))))
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) * (1.0 + wj))) + (fma(wj, wj, pow(wj, 4.0)) - pow(wj, 3.0));
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target12.9
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. Simplified12.9

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

  3. Applied *-un-lft-identity_binary6412.9

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

  4. Applied *-un-lft-identity_binary6412.9

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

  5. Applied distribute-lft-out_binary6412.9

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

  6. Simplified6.8

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

  7. Taylor expanded in wj around 0 1.2

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

  8. Simplified1.2

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

  9. Final simplification1.2

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

Reproduce

herbie shell --seed 2021357 
(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))))))