Average Error: 13.7 → 0.7
Time: 54.9s
Precision: 64
Internal Precision: 128
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 7.939557038379471 \cdot 10^{-18}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\left(wj - \frac{1}{e^{wj}} \cdot x\right) \cdot \left(wj \cdot wj + \left(1 - wj\right)\right)\right) \cdot \frac{1}{(\left(wj \cdot wj\right) \cdot wj + 1)_*}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.0
Herbie0.7
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 7.939557038379471e-18

    1. Initial program 18.1

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.7

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

    if 7.939557038379471e-18 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

    1. Initial program 3.1

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

    3. Applied distribute-rgt1-in3.1

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

    4. Applied *-un-lft-identity3.1

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

    5. Applied times-frac3.2

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

    6. Simplified0.6

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

    8. Applied flip3-+0.6

      \[\leadsto wj - \frac{1}{\color{blue}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)}}} \cdot \left(wj - \frac{x}{e^{wj}}\right)\]

    9. Applied associate-/r/0.6

      \[\leadsto wj - \color{blue}{\left(\frac{1}{{wj}^{3} + {1}^{3}} \cdot \left(wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)\right)\right)} \cdot \left(wj - \frac{x}{e^{wj}}\right)\]

    10. Applied associate-*l*0.6

      \[\leadsto wj - \color{blue}{\frac{1}{{wj}^{3} + {1}^{3}} \cdot \left(\left(wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)\right) \cdot \left(wj - \frac{x}{e^{wj}}\right)\right)}\]

    11. Simplified0.6

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

    13. Applied div-inv0.6

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 7.939557038379471 \cdot 10^{-18}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\left(wj - \frac{1}{e^{wj}} \cdot x\right) \cdot \left(wj \cdot wj + \left(1 - wj\right)\right)\right) \cdot \frac{1}{(\left(wj \cdot wj\right) \cdot wj + 1)_*}\\ \end{array}\]

Runtime

Time bar (total: 54.9s)Debug logProfile

herbie shell --seed 2018278 +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))))))