Error

Target

Derivation

1. Split input into 2 regimes

2. if wj < 9.312929706283876e-05

   1. Initial program 13.1

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

   3. Applied div-sub13.1

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

   4. Applied associate--r-6.8

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

   5. Simplified6.8

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

   6. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}\]

   7. Simplified0.3

      \[\leadsto \color{blue}{(\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + \left({wj}^{4}\right))_*} + \frac{x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}\]
   8. Using strategy rm

   9. Applied fma-udef0.3

      \[\leadsto \color{blue}{\left(\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + {wj}^{4}\right)} + \frac{x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}\]

   10. Applied associate-+l+0.3

       \[\leadsto \color{blue}{\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + \left({wj}^{4} + \frac{x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}\right)}\]

   if 9.312929706283876e-05 < wj

   1. Initial program 32.4

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

   3. Applied div-sub32.5

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

   4. Applied associate--r-32.5

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

   5. Simplified32.4

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

   7. Applied *-un-lft-identity32.4

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

   8. Applied times-frac32.4

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

   9. Simplified32.4

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

   10. Simplified1.1

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.312929706283876 \cdot 10^{-05}:\\ \;\;\;\;\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + \left(\frac{x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*} + {wj}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - wj \cdot \frac{1}{1 + wj}\right) + \frac{x}{(wj \cdot \left(e^{wj}\right) + \left(e^{wj}\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 14.9s)Debug logProfile

herbie shell --seed 2018290 +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))))))