Average Error: 13.6 → 0.4
Time: 55.0s
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)} \cdot \sqrt[3]{\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)}\right) \cdot \sqrt[3]{\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)} \le -5.946392597907632 \cdot 10^{+19}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{elif}\;\left(\sqrt[3]{\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)} \cdot \sqrt[3]{\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)}\right) \cdot \sqrt[3]{\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)} \le 3.568793011148332 \cdot 10^{-19}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

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Results

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Target

Original13.6
Target13.0
Herbie0.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* (cbrt (- (+ (pow wj 2) x) (* 2 (* wj x)))) (cbrt (- (+ (pow wj 2) x) (* 2 (* wj x))))) (cbrt (- (+ (pow wj 2) x) (* 2 (* wj x))))) < -5.946392597907632e+19 or 3.568793011148332e-19 < (* (* (cbrt (- (+ (pow wj 2) x) (* 2 (* wj x)))) (cbrt (- (+ (pow wj 2) x) (* 2 (* wj x))))) (cbrt (- (+ (pow wj 2) x) (* 2 (* wj x)))))

    1. Initial program 1.5

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

    3. Applied div-sub1.5

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

    4. Applied associate--r-1.5

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

    5. Applied simplify0.4

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

    if -5.946392597907632e+19 < (* (* (cbrt (- (+ (pow wj 2) x) (* 2 (* wj x)))) (cbrt (- (+ (pow wj 2) x) (* 2 (* wj x))))) (cbrt (- (+ (pow wj 2) x) (* 2 (* wj x))))) < 3.568793011148332e-19

    1. Initial program 26.5

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

    2. Taylor expanded around 0 26.5

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

    3. Applied simplify0.3

      \[\leadsto \color{blue}{x + wj \cdot wj}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 55.0s)Debug logProfile

herbie shell --seed 2018208 
(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))))))