Average Error: 13.7 → 1.1
Time: 4.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(\sqrt{{wj}^{4} + {wj}^{2}}, \sqrt{{wj}^{4} + {wj}^{2}}, -{wj}^{3}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(\sqrt{{wj}^{4} + {wj}^{2}}, \sqrt{{wj}^{4} + {wj}^{2}}, -{wj}^{3}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}
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(sqrt((pow(wj, 4.0) + pow(wj, 2.0))), sqrt((pow(wj, 4.0) + pow(wj, 2.0))), -pow(wj, 3.0)) + (x / (exp(wj) + (wj * exp(wj)))));
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.2
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
  2. Using strategy rm

  3. Applied div-sub13.7

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

  4. Applied associate--r-7.4

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

  5. Simplified7.4

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

  6. Taylor expanded around 0 1.1

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

  8. Applied add-sqr-sqrt1.1

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

  9. Applied fma-neg1.1

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

  10. Final simplification1.1

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

Reproduce

herbie shell --seed 2020106 +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))))))