Average Error: 13.8 → 0.5
Time: 1.2m
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left((\left(x \cdot 2\right) \cdot \left(-wj\right) + \left((wj \cdot wj + x)_*\right))_*\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)} \cdot \sqrt[3]{(\left(x \cdot 2\right) \cdot \left(-wj\right) + \left((wj \cdot wj + x)_*\right))_*} \le 1.1418251620986433 \cdot 10^{-15}:\\ \;\;\;\;(\left(x \cdot 2\right) \cdot \left(-wj\right) + \left((wj \cdot wj + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.2
Herbie0.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* (pow (fma (* x 2) (- wj) (fma wj wj x)) (+ 1/3 1/3)) (cbrt (fma (* x 2) (- wj) (fma wj wj x)))) < 1.1418251620986433e-15

    1. Initial program 34.7

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

    2. Taylor expanded around 0 0.6

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

    3. Applied simplify0.6

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

    if 1.1418251620986433e-15 < (* (pow (fma (* x 2) (- wj) (fma wj wj x)) (+ 1/3 1/3)) (cbrt (fma (* x 2) (- wj) (fma wj wj x))))

    1. Initial program 5.6

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

    3. Applied div-sub5.6

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

    4. Applied associate--r-1.3

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

    5. Applied simplify0.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +o rules:numerics
(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))))))