Average Error: 13.6 → 0.8
Time: 39.4s
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right) \le 1.1273399280310297 \cdot 10^{-25}:\\ \;\;\;\;\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{wj - \frac{wj}{1 + wj}} \cdot \sqrt[3]{wj - \frac{wj}{1 + wj}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{wj - \frac{wj}{1 + wj}}} \cdot \sqrt[3]{\sqrt[3]{wj - \frac{wj}{1 + wj}}}\right) \cdot \sqrt[3]{\sqrt[3]{wj - \frac{wj}{1 + wj}}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.6
Target13.1
Herbie0.8
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- (+ (pow wj 2) x) (* 2 (* wj x))) < 1.1273399280310297e-25

    1. Initial program 18.2

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

    2. Taylor expanded around 0 0.6

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

    if 1.1273399280310297e-25 < (- (+ (pow wj 2) x) (* 2 (* wj x)))

    1. Initial program 2.5

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

    3. Applied div-sub2.5

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

    4. Applied associate--r-2.5

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

    5. Applied simplify1.3

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

    7. Applied add-cube-cbrt1.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{wj - \frac{wj}{1 + wj}} \cdot \sqrt[3]{wj - \frac{wj}{1 + wj}}\right) \cdot \sqrt[3]{wj - \frac{wj}{1 + wj}}} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt1.3

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

Runtime

Time bar (total: 39.4s)Debug logProfile

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