Average Error: 13.6 → 1.7
Time: 7.2s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left(2, x \cdot wj, \mathsf{fma}\left(2.6666666666666665, x \cdot {wj}^{3}, {wj}^{3}\right)\right) \]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left(2, x \cdot wj, \mathsf{fma}\left(2.6666666666666665, x \cdot {wj}^{3}, {wj}^{3}\right)\right)

(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (-
  (fma 2.5 (* x (* wj wj)) (fma wj wj x))
  (fma 2.0 (* x wj) (fma 2.6666666666666665 (* x (pow wj 3.0)) (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 fma(2.5, (x * (wj * wj)), fma(wj, wj, x)) - fma(2.0, (x * wj), fma(2.6666666666666665, (x * pow(wj, 3.0)), pow(wj, 3.0)));
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.6
Target13.0
Herbie1.7
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Initial program 13.6

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

  2. Taylor expanded in wj around 0 1.7

    \[\leadsto \color{blue}{\left(2.5 \cdot \left({wj}^{2} \cdot x\right) + \left({wj}^{2} + x\right)\right) - \left(2 \cdot \left(wj \cdot x\right) + \left(2.6666666666666665 \cdot \left({wj}^{3} \cdot x\right) + {wj}^{3}\right)\right)} \]

  3. Simplified1.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left(2, x \cdot wj, \mathsf{fma}\left(2.6666666666666665, x \cdot {wj}^{3}, {wj}^{3}\right)\right)} \]

  4. Final simplification1.7

    \[\leadsto \mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left(2, x \cdot wj, \mathsf{fma}\left(2.6666666666666665, x \cdot {wj}^{3}, {wj}^{3}\right)\right) \]

Reproduce

herbie shell --seed 2021313 
(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))))))