Average Error: 13.6 → 1.1

Time: 26.7s

Precision: 64

Internal Precision: 832

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `wj`

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Target

Original	13.6
Target	13.1
Herbie	1.1

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

Derivation

Initial program 13.6
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Initial simplification6.7
\[\leadsto \left(wj - \frac{wj}{wj + 1}\right) + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\]
Taylor expanded around 0 1.1
\[\leadsto \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)} + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\]
Using strategy rm
Applied add-cube-cbrt1.1
\[\leadsto \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{x}{e^{wj} \cdot \color{blue}{\left(\left(\sqrt[3]{wj + 1} \cdot \sqrt[3]{wj + 1}\right) \cdot \sqrt[3]{wj + 1}\right)}}\]
Applied associate-*r*1.1
\[\leadsto \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{x}{\color{blue}{\left(e^{wj} \cdot \left(\sqrt[3]{wj + 1} \cdot \sqrt[3]{wj + 1}\right)\right) \cdot \sqrt[3]{wj + 1}}}\]
Final simplification1.1
\[\leadsto \frac{x}{\left(e^{wj} \cdot \left(\sqrt[3]{wj + 1} \cdot \sqrt[3]{wj + 1}\right)\right) \cdot \sqrt[3]{wj + 1}} + \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)\]

Runtime

herbie shell --seed 2018215 
(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))))))