Average Error: 13.7 → 2.2
Time: 4.9s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[1 \cdot \mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
1 \cdot \mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\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 ((1.0 * fma(wj, wj, x)) - (2.0 * (wj * x)));
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.0
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

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

  2. Simplified13.0

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

  3. Taylor expanded around 0 2.2

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

  5. Applied *-un-lft-identity2.2

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

  6. Applied *-un-lft-identity2.2

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

  7. Applied distribute-lft-out2.2

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

  8. Simplified2.2

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(wj, wj, x\right)} - 2 \cdot \left(wj \cdot x\right)\]

  9. Final simplification2.2

    \[\leadsto 1 \cdot \mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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))))))