Average Error: 13.6 → 0.6
Time: 59.9s
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt{{\left(\frac{1}{wj}\right)}^{-2}} \cdot \sqrt{{\left(\frac{1}{wj}\right)}^{-2}} + x\right) - 2 \cdot \left(wj \cdot x\right) \le 2.8784524307538913 \cdot 10^{-16}:\\ \;\;\;\;\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (- (+ (* (sqrt (pow (/ 1 wj) -2)) (sqrt (pow (/ 1 wj) -2))) x) (* 2 (* wj x))) < 2.8784524307538913e-16

    1. Initial program 18.0

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

    2. Taylor expanded around 0 0.7

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

    if 2.8784524307538913e-16 < (- (+ (* (sqrt (pow (/ 1 wj) -2)) (sqrt (pow (/ 1 wj) -2))) x) (* 2 (* wj x)))

    1. Initial program 2.1

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

    3. Applied div-sub2.1

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

    4. Applied associate--r-2.1

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

    5. Applied simplify0.5

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

Runtime

Time bar (total: 59.9s)Debug logProfile

herbie shell --seed 2018207 
(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))))))