Average Error: 13.4 → 1.2
Time: 58.5s
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{\frac{x}{e^{wj}}}{\sqrt{1 + wj}}}{\sqrt{1 + wj}}\]

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original13.4
Target12.9
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

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

  2. Initial simplification6.5

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

  3. Taylor expanded around 0 1.1

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

  5. Applied add-sqr-sqrt1.2

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

  6. Applied associate-/r*1.2

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

  7. Final simplification1.2

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

Runtime

Time bar (total: 58.5s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))