Average Error: 13.5 → 1.0

Time: 20.8s

Precision: 64

Internal Precision: 832

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

↓

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

Error

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

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	13.5
Target	12.9
Herbie	1.0

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

Derivation

Initial program 13.5
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Initial simplification6.8
\[\leadsto \left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
Taylor expanded around 0 1.0
\[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
Using strategy rm
Applied associate-/l/1.0
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\frac{x}{\left(wj + 1\right) \cdot e^{wj}}}\]
Final simplification1.0
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{x}{\left(wj + 1\right) \cdot e^{wj}}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	1.0	1.0	0.1	1.0	0%

herbie shell --seed 2018295 
(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))))))