Average Error: 13.6 → 0.9
Time: 1.1m
Precision: 64
Internal Precision: 896
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{wj} + wj\right) - (\left(\frac{1}{wj}\right) \cdot \left(\frac{1}{wj}\right) + 1)_* \le -134303740816757.34:\\ \;\;\;\;(\left(x \cdot 2\right) \cdot \left(-wj\right) + \left((wj \cdot wj + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{\frac{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}{(\left(e^{wj}\right) \cdot wj + \left(-x\right))_*}}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.6
Target13.0
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- (+ (/ 1 wj) wj) (fma (/ 1 wj) (/ 1 wj) 1)) < -134303740816757.34

    1. Initial program 13.5

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

    2. Applied simplify13.5

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

    3. Taylor expanded around 0 0.2

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

    4. Applied simplify0.2

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

    if -134303740816757.34 < (- (+ (/ 1 wj) wj) (fma (/ 1 wj) (/ 1 wj) 1))

    1. Initial program 18.1

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

    2. Applied simplify18.1

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

    4. Applied clear-num18.1

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +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))))))