Average Error: 13.5 → 0.2
Time: 1.0m
Precision: 64
Internal Precision: 896
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \frac{{\left({wj}^{4}\right)}^{3} + {\left(\left(1 - wj\right) \cdot \left(wj \cdot wj\right)\right)}^{3}}{\left(\left(1 - wj\right) \cdot \left(wj \cdot wj\right)\right) \cdot \left(\left(1 - wj\right) \cdot \left(wj \cdot wj\right) - {wj}^{4}\right) + {wj}^{4} \cdot {wj}^{4}} \le 3.2733933496132436 \cdot 10^{-07}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \left({wj}^{4} + \left(1 - wj\right) \cdot \left(wj \cdot wj\right)\right)\\ \mathbf{if}\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \frac{{\left({wj}^{4}\right)}^{3} + {\left(\left(1 - wj\right) \cdot \left(wj \cdot wj\right)\right)}^{3}}{\left(\left(1 - wj\right) \cdot \left(wj \cdot wj\right)\right) \cdot \left(\left(1 - wj\right) \cdot \left(wj \cdot wj\right) - {wj}^{4}\right) + {wj}^{4} \cdot {wj}^{4}} \le 3.5891781514450206 \cdot 10^{+85}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target13.0
Herbie0.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (+ (/ (/ x (exp wj)) (+ 1 wj)) (/ (+ (pow (pow wj 4) 3) (pow (* (- 1 wj) (* wj wj)) 3)) (+ (* (* (- 1 wj) (* wj wj)) (- (* (- 1 wj) (* wj wj)) (pow wj 4))) (* (pow wj 4) (pow wj 4))))) < 3.2733933496132436e-07

    1. Initial program 31.1

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

    3. Applied div-sub31.1

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

    4. Applied associate--r-27.5

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

    5. Applied simplify27.4

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

    6. Taylor expanded around 0 0.3

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

    7. Applied simplify0.3

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

    if 3.2733933496132436e-07 < (+ (/ (/ x (exp wj)) (+ 1 wj)) (/ (+ (pow (pow wj 4) 3) (pow (* (- 1 wj) (* wj wj)) 3)) (+ (* (* (- 1 wj) (* wj wj)) (- (* (- 1 wj) (* wj wj)) (pow wj 4))) (* (pow wj 4) (pow wj 4))))) < 3.5891781514450206e+85

    1. Initial program 14.5

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

    3. Applied div-sub14.5

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

    4. Applied simplify0.4

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

    5. Applied simplify0.4

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

    if 3.5891781514450206e+85 < (+ (/ (/ x (exp wj)) (+ 1 wj)) (/ (+ (pow (pow wj 4) 3) (pow (* (- 1 wj) (* wj wj)) 3)) (+ (* (* (- 1 wj) (* wj wj)) (- (* (- 1 wj) (* wj wj)) (pow wj 4))) (* (pow wj 4) (pow wj 4)))))

    1. Initial program 9.2

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

    2. Taylor expanded around 0 0.2

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(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))))))