Average Error: 13.6 → 1.1
Time: 1.2m
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\log \left(wj - \frac{(\left(e^{wj}\right) \cdot wj + \left(-x\right))_*}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}\right)} \le 1.6447664544199101 \cdot 10^{-18}:\\ \;\;\;\;(\left(x \cdot 2\right) \cdot \left(-wj\right) + \left((wj \cdot wj + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;wj - (\left(e^{wj}\right) \cdot wj + \left(-x\right))_* \cdot \frac{1}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.6
Target13.1
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 (log (- wj (/ (fma (exp wj) wj (- x)) (fma wj (exp wj) (exp wj)))))) < 1.6447664544199101e-18

    1. Initial program 38.0

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

    2. Applied simplify38.0

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

    3. Taylor expanded around 0 0.3

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

    4. Applied simplify0.3

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

    if 1.6447664544199101e-18 < (exp (log (- wj (/ (fma (exp wj) wj (- x)) (fma wj (exp wj) (exp wj))))))

    1. Initial program 1.5

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

    2. Applied simplify1.5

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

    4. Applied div-inv1.5

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +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))))))