Jmat.Real.lambertw, newton loop step

Average Error: 13.3 → 1.3

Time: 4.4s

Precision: binary64

• could not determine a ground truth for program body

  1. wj = -8.761369592446147e+49
  2. x = -7.931834300016019e-284

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le 3.03963131152792666 \cdot 10^{-12}:\\ \;\;\;\;x + {\left(wj \cdot wj\right)}^{\frac{1}{3}} \cdot \left(\left(wj + x \cdot -2\right) \cdot \sqrt[3]{wj}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original	13.3
Target	12.6
Herbie	1.3

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

Derivation

1. Split input into 2 regimes

2. if wj < 3.03963131152792666e-12

   1. Initial program 12.9

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

   2. Simplified 12.9

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}}\]

   3. Taylor expanded around 0 0.7

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

   4. Simplified 0.8

      \[\leadsto \color{blue}{x + wj \cdot \left(wj - x \cdot 2\right)}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 0.9

      \[\leadsto x + \color{blue}{\left(\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right) \cdot \sqrt[3]{wj}\right)} \cdot \left(wj - x \cdot 2\right)\]

   7. Applied associate-*l* 0.9

      \[\leadsto x + \color{blue}{\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right) \cdot \left(\sqrt[3]{wj} \cdot \left(wj - x \cdot 2\right)\right)}\]

   8. Simplified 0.9

      \[\leadsto x + \left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right) \cdot \color{blue}{\left(\left(wj + x \cdot -2\right) \cdot \sqrt[3]{wj}\right)}\]
   9. Using strategy rm

   10. Applied pow1/3 32.7

       \[\leadsto x + \left(\sqrt[3]{wj} \cdot \color{blue}{{wj}^{\frac{1}{3}}}\right) \cdot \left(\left(wj + x \cdot -2\right) \cdot \sqrt[3]{wj}\right)\]

   11. Applied pow1/3 32.7

       \[\leadsto x + \left(\color{blue}{{wj}^{\frac{1}{3}}} \cdot {wj}^{\frac{1}{3}}\right) \cdot \left(\left(wj + x \cdot -2\right) \cdot \sqrt[3]{wj}\right)\]

   12. Applied pow-prod-down 1.2

       \[\leadsto x + \color{blue}{{\left(wj \cdot wj\right)}^{\frac{1}{3}}} \cdot \left(\left(wj + x \cdot -2\right) \cdot \sqrt[3]{wj}\right)\]

   if 3.03963131152792666e-12 < wj

   1. Initial program 25.0

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

   2. Simplified 4.1

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}}\]
   3. Using strategy rm

   4. Applied div-inv 4.1

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

   5. Simplified 4.0

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 3.03963131152792666 \cdot 10^{-12}:\\ \;\;\;\;x + {\left(wj \cdot wj\right)}^{\frac{1}{3}} \cdot \left(\left(wj + x \cdot -2\right) \cdot \sqrt[3]{wj}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))