Average Error: 13.6 → 1.6
Time: 1.0m
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{{wj}^{2} + x} \cdot \sqrt[3]{{wj}^{2} + x}\right) \cdot \left(\sqrt[3]{\sqrt{{wj}^{2} + x}} \cdot \sqrt[3]{\sqrt{{wj}^{2} + x}}\right) - 2 \cdot \left(wj \cdot x\right) \le 6.588975398815594 \cdot 10^{-26}:\\ \;\;\;\;\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{if}\;\left(\sqrt[3]{{wj}^{2} + x} \cdot \sqrt[3]{{wj}^{2} + x}\right) \cdot \left(\sqrt[3]{\sqrt{{wj}^{2} + x}} \cdot \sqrt[3]{\sqrt{{wj}^{2} + x}}\right) - 2 \cdot \left(wj \cdot x\right) \le 1.2942413642409161 \cdot 10^{+130}:\\ \;\;\;\;\frac{wj \cdot wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}{wj + \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\\ \mathbf{else}:\\ \;\;\;\;\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.6
Target13.0
Herbie1.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* (* (cbrt (+ (pow wj 2) x)) (cbrt (+ (pow wj 2) x))) (* (cbrt (sqrt (+ (pow wj 2) x))) (cbrt (sqrt (+ (pow wj 2) x))))) (* 2 (* wj x))) < 6.588975398815594e-26 or 1.2942413642409161e+130 < (- (* (* (cbrt (+ (pow wj 2) x)) (cbrt (+ (pow wj 2) x))) (* (cbrt (sqrt (+ (pow wj 2) x))) (cbrt (sqrt (+ (pow wj 2) x))))) (* 2 (* wj x)))

    1. Initial program 15.2

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

    2. Taylor expanded around 0 1.1

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

    if 6.588975398815594e-26 < (- (* (* (cbrt (+ (pow wj 2) x)) (cbrt (+ (pow wj 2) x))) (* (cbrt (sqrt (+ (pow wj 2) x))) (cbrt (sqrt (+ (pow wj 2) x))))) (* 2 (* wj x))) < 1.2942413642409161e+130

    1. Initial program 4.7

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

    3. Applied flip--4.8

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' 
(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))))))