Average Error: 14.6 → 1.2

Time: 33.3s

Precision: 64

Internal Precision: 128

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

↓

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

Error

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

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Target

Original	14.6
Target	13.9
Herbie	1.2

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

Derivation

Initial program 14.6
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Initial simplification7.4
\[\leadsto \left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
Taylor expanded around 0 1.2
\[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
Using strategy rm
Applied add-cube-cbrt1.2
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{\color{blue}{\left(\sqrt[3]{e^{wj}} \cdot \sqrt[3]{e^{wj}}\right) \cdot \sqrt[3]{e^{wj}}}}}{wj + 1}\]
Applied associate-/r*1.2
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{\color{blue}{\frac{\frac{x}{\sqrt[3]{e^{wj}} \cdot \sqrt[3]{e^{wj}}}}{\sqrt[3]{e^{wj}}}}}{wj + 1}\]
Final simplification1.2
\[\leadsto \frac{\frac{\frac{x}{\sqrt[3]{e^{wj}} \cdot \sqrt[3]{e^{wj}}}}{\sqrt[3]{e^{wj}}}}{1 + wj} + \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	1.2	1.2	0.1	1.1	0%

herbie shell --seed 2018355 
(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))))))