Average Error: 13.3 → 1.1
Time: 1.9m
Precision: 64
Internal Precision: 128
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{wj} + wj \cdot e^{wj} \le 1.0000000002915277:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(wj \cdot wj + \left(\left(\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right) \cdot \left(\left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\right)\right))_*}{\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right) + wj}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.3
Target12.7
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (exp wj) (* wj (exp wj))) < 1.0000000002915277

    1. Initial program 13.0

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.8

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

    if 1.0000000002915277 < (+ (exp wj) (* wj (exp wj)))

    1. Initial program 25.8

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    2. Using strategy rm

    3. Applied distribute-rgt1-in25.9

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

    4. Applied *-un-lft-identity25.9

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

    5. Applied times-frac25.8

      \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}}\]

    6. Simplified3.4

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

    8. Applied flip--12.6

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

    10. Applied fma-neg12.6

      \[\leadsto \frac{\color{blue}{(wj \cdot wj + \left(-\left(\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right) \cdot \left(\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right)\right))_*}}{wj + \frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{wj} + wj \cdot e^{wj} \le 1.0000000002915277:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(wj \cdot wj + \left(\left(\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\right) \cdot \left(\left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\right)\right))_*}{\frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right) + wj}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018274 +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))))))