Jmat.Real.lambertw, newton loop step

Average Error: 13.5 → 0.9

Time: 6.1s

Precision: 64

• could not determine a ground truth for program body

  1. wj = 8.677094401051118e+132
  2. x = 1.662100557100748e-24

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le 1.3121422791952024 \cdot 10^{-8}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left(\frac{\frac{x}{wj \cdot wj - 1}}{\frac{e^{wj}}{wj - 1}} + wj\right) - \frac{\frac{wj}{\sqrt{wj + 1}}}{\sqrt{wj + 1}}} \cdot \sqrt[3]{\left(\frac{\frac{x}{wj \cdot wj - 1}}{\frac{e^{wj}}{wj - 1}} + wj\right) - \frac{\frac{wj}{\sqrt{wj + 1}}}{\sqrt{wj + 1}}}\right) \cdot \sqrt[3]{\left(\frac{\frac{x}{wj \cdot wj - 1}}{\frac{e^{wj}}{wj - 1}} + wj\right) - \frac{\frac{wj}{\sqrt{wj + 1}}}{\sqrt{wj + 1}}}\\ \end{array}\]

C

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

↓

double code(double wj, double x) {
	double VAR;
	if ((wj <= 1.3121422791952024e-08)) {
		VAR = ((x + pow(wj, 2.0)) - (2.0 * (wj * x)));
	} else {
		VAR = ((cbrt(((((x / ((wj * wj) - 1.0)) / (exp(wj) / (wj - 1.0))) + wj) - ((wj / sqrt((wj + 1.0))) / sqrt((wj + 1.0))))) * cbrt(((((x / ((wj * wj) - 1.0)) / (exp(wj) / (wj - 1.0))) + wj) - ((wj / sqrt((wj + 1.0))) / sqrt((wj + 1.0)))))) * cbrt(((((x / ((wj * wj) - 1.0)) / (exp(wj) / (wj - 1.0))) + wj) - ((wj / sqrt((wj + 1.0))) / sqrt((wj + 1.0))))));
	}
	return VAR;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.5
Target	12.7
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if wj < 1.3121422791952024e-08

   1. Initial program 13.0

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

   2. Simplified 13.0

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

   3. Taylor expanded around 0 0.8

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

   if 1.3121422791952024e-08 < wj

   1. Initial program 30.9

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

   2. Simplified 3.3

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

   4. Applied flip-+ 3.3

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

   5. Applied associate-/r/ 3.3

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

   6. Applied associate-/l* 3.4

      \[\leadsto \left(\color{blue}{\frac{\frac{x}{wj \cdot wj - 1 \cdot 1}}{\frac{e^{wj}}{wj - 1}}} + wj\right) - \frac{wj}{wj + 1}\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 3.6

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

   9. Applied associate-/r* 3.6

      \[\leadsto \left(\frac{\frac{x}{wj \cdot wj - 1 \cdot 1}}{\frac{e^{wj}}{wj - 1}} + wj\right) - \color{blue}{\frac{\frac{wj}{\sqrt{wj + 1}}}{\sqrt{wj + 1}}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 4.5

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

   12. Simplified 4.5

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

   13. Simplified 4.5

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.3121422791952024 \cdot 10^{-8}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left(\frac{\frac{x}{wj \cdot wj - 1}}{\frac{e^{wj}}{wj - 1}} + wj\right) - \frac{\frac{wj}{\sqrt{wj + 1}}}{\sqrt{wj + 1}}} \cdot \sqrt[3]{\left(\frac{\frac{x}{wj \cdot wj - 1}}{\frac{e^{wj}}{wj - 1}} + wj\right) - \frac{\frac{wj}{\sqrt{wj + 1}}}{\sqrt{wj + 1}}}\right) \cdot \sqrt[3]{\left(\frac{\frac{x}{wj \cdot wj - 1}}{\frac{e^{wj}}{wj - 1}} + wj\right) - \frac{\frac{wj}{\sqrt{wj + 1}}}{\sqrt{wj + 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

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