Average Error: 13.7 → 2.4
Time: 1.2m
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[(wj \cdot wj + x)_*\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.1
Herbie2.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

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

  2. Taylor expanded around 0 15.1

    \[\leadsto wj - \color{blue}{\left(wj - \left({wj}^{2} + x\right)\right)}\]

  3. Applied simplify2.4

    \[\leadsto \color{blue}{(wj \cdot wj + x)_*}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(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))))))