Jmat.Real.lambertw, newton loop step

Average Error: 13.3 → 1.4

Time: 9.3s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le -3.868446554170448 \cdot 10^{-310}:\\ \;\;\;\;x + \left({\left(\sqrt[3]{-1}\right)}^{2} \cdot {\left(\frac{-1}{wj}\right)}^{\frac{-2}{3}}\right) \cdot \left(\sqrt[3]{wj} \cdot \left(wj - 2 \cdot x\right)\right)\\ \mathbf{elif}\;wj \le 9.93050096655964495 \cdot 10^{-9}:\\ \;\;\;\;x + \left({\left({\left(\sqrt[3]{\sqrt[3]{wj}}\right)}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{4}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{wj}} \cdot \sqrt[3]{\sqrt[3]{wj}}\right)\right) \cdot \left(\sqrt[3]{wj} \cdot \left(wj - 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

C

double code(double wj, double x) {
	return ((double) (wj - (((double) (((double) (wj * ((double) exp(wj)))) - x)) / ((double) (((double) exp(wj)) + ((double) (wj * ((double) exp(wj)))))))));
}

↓

double code(double wj, double x) {
	double VAR;
	if ((wj <= -3.86844655417045e-310)) {
		VAR = ((double) (x + ((double) (((double) (((double) pow(((double) cbrt(-1.0)), 2.0)) * ((double) pow((-1.0 / wj), -0.6666666666666666)))) * ((double) (((double) cbrt(wj)) * ((double) (wj - ((double) (2.0 * x))))))))));
	} else {
		double VAR_1;
		if ((wj <= 9.930500966559645e-09)) {
			VAR_1 = ((double) (x + ((double) (((double) (((double) pow(((double) pow(((double) cbrt(((double) cbrt(wj)))), ((double) (((double) cbrt(4.0)) * ((double) cbrt(4.0)))))), ((double) cbrt(4.0)))) * ((double) (((double) cbrt(((double) cbrt(wj)))) * ((double) cbrt(((double) cbrt(wj)))))))) * ((double) (((double) cbrt(wj)) * ((double) (wj - ((double) (2.0 * x))))))))));
		} else {
			VAR_1 = ((double) (((double) (wj - (wj / ((double) (wj + 1.0))))) + ((x / ((double) exp(wj))) / ((double) (wj + 1.0)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.3
Target	12.6
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if wj < -3.868446554170448e-310

   1. Initial program 13.2

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

   2. Simplified 13.2

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

   3. Taylor expanded around 0 1.3

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

   4. Simplified 1.4

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

   6. Applied add-cube-cbrt 1.5

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

   7. Applied associate-*l* 1.6

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

   8. Taylor expanded around -inf 1.9

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

   9. Simplified 1.9

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

   if -3.868446554170448e-310 < wj < 9.93050096655964495e-9

   1. Initial program 12.6

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

   2. Simplified 12.6

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

   3. Taylor expanded around 0 0.1

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

   4. Simplified 0.2

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

   6. Applied add-cube-cbrt 0.3

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

   7. Applied associate-*l* 0.3

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

   9. Applied add-cube-cbrt 0.4

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

   10. Applied add-cube-cbrt 0.4

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

   11. Applied swap-sqr 0.4

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

   12. Simplified 0.4

       \[\leadsto x + \left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{wj}}\right)}^{4}} \cdot \left(\sqrt[3]{\sqrt[3]{wj}} \cdot \sqrt[3]{\sqrt[3]{wj}}\right)\right) \cdot \left(\sqrt[3]{wj} \cdot \left(wj - 2 \cdot x\right)\right)\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 0.8

       \[\leadsto x + \left({\left(\sqrt[3]{\sqrt[3]{wj}}\right)}^{\color{blue}{\left(\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right) \cdot \sqrt[3]{4}\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{wj}} \cdot \sqrt[3]{\sqrt[3]{wj}}\right)\right) \cdot \left(\sqrt[3]{wj} \cdot \left(wj - 2 \cdot x\right)\right)\]

   15. Applied pow-unpow 0.8

       \[\leadsto x + \left(\color{blue}{{\left({\left(\sqrt[3]{\sqrt[3]{wj}}\right)}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{4}\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{wj}} \cdot \sqrt[3]{\sqrt[3]{wj}}\right)\right) \cdot \left(\sqrt[3]{wj} \cdot \left(wj - 2 \cdot x\right)\right)\]

   if 9.93050096655964495e-9 < wj

   1. Initial program 27.2

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

   2. Simplified 2.5

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

   4. Applied div-sub 2.5

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

   5. Applied associate--r- 2.5

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -3.868446554170448 \cdot 10^{-310}:\\ \;\;\;\;x + \left({\left(\sqrt[3]{-1}\right)}^{2} \cdot {\left(\frac{-1}{wj}\right)}^{\frac{-2}{3}}\right) \cdot \left(\sqrt[3]{wj} \cdot \left(wj - 2 \cdot x\right)\right)\\ \mathbf{elif}\;wj \le 9.93050096655964495 \cdot 10^{-9}:\\ \;\;\;\;x + \left({\left({\left(\sqrt[3]{\sqrt[3]{wj}}\right)}^{\left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right)}\right)}^{\left(\sqrt[3]{4}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{wj}} \cdot \sqrt[3]{\sqrt[3]{wj}}\right)\right) \cdot \left(\sqrt[3]{wj} \cdot \left(wj - 2 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

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))))))