Jmat.Real.lambertw, newton loop step

Average Error: 13.9 → 0.9

Time: 5.9s

Precision: 64

• could not determine a ground truth for program body

  1. wj = -1.3277474740950274e+60
  2. x = 5.872367900798126e-198

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le 4.90683548158770919 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{x}{{\left(wj \cdot wj\right)}^{3} - 1} \cdot \left(\left(wj - 1\right) \cdot {wj}^{4} + \left(wj - 1\right) \cdot \mathsf{fma}\left(wj, wj, 1\right)\right)}{e^{wj}} + wj\right) - \frac{\mathsf{fma}\left(wj, wj, 1 - wj\right) \cdot wj}{{wj}^{3} + 1}\right) + \mathsf{fma}\left(wj, wj, 1 - wj\right) \cdot \left(\left(-\frac{wj}{{wj}^{3} + 1}\right) + \frac{wj}{{wj}^{3} + 1}\right)\\ \end{array}\]

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 temp;
	if ((wj <= 4.906835481587709e-09)) {
		temp = ((x + pow(wj, 2.0)) - (2.0 * (wj * x)));
	} else {
		temp = ((((((x / (pow((wj * wj), 3.0) - 1.0)) * (((wj - 1.0) * pow(wj, 4.0)) + ((wj - 1.0) * fma(wj, wj, 1.0)))) / exp(wj)) + wj) - ((fma(wj, wj, (1.0 - wj)) * wj) / (pow(wj, 3.0) + 1.0))) + (fma(wj, wj, (1.0 - wj)) * (-(wj / (pow(wj, 3.0) + 1.0)) + (wj / (pow(wj, 3.0) + 1.0)))));
	}
	return temp;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.9
Target	13.4
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 < 4.906835481587709e-09

   1. Initial program 13.7

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

   2. Simplified 13.6

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

   3. Taylor expanded around 0 0.8

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

   if 4.906835481587709e-09 < wj

   1. Initial program 24.1

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

   2. Simplified 3.8

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

   4. Applied flip3-+ 3.9

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

   5. Applied associate-/r/ 3.8

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

   6. Applied add-sqr-sqrt 15.6

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

   7. Applied prod-diff 15.6

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

   8. Simplified 3.8

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

   9. Simplified 3.9

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

   11. Applied flip-+ 4.0

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

   12. Applied associate-/r/ 3.9

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

   13. Simplified 3.9

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

   15. Applied flip3-- 4.0

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

   16. Applied associate-/r/ 4.0

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

   17. Applied associate-*l* 4.0

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

   18. Simplified 4.0

       \[\leadsto \left(\left(\frac{\frac{x}{{\left(wj \cdot wj\right)}^{3} - {1}^{3}} \cdot \color{blue}{\left(\left(wj - 1\right) \cdot {wj}^{4} + \left(wj - 1\right) \cdot \mathsf{fma}\left(wj, wj, 1\right)\right)}}{e^{wj}} + wj\right) - \frac{\mathsf{fma}\left(wj, wj, 1 - wj\right) \cdot wj}{{wj}^{3} + 1}\right) + \mathsf{fma}\left(wj, wj, 1 - wj\right) \cdot \left(\left(-\frac{wj}{{wj}^{3} + 1}\right) + \frac{wj}{{wj}^{3} + 1}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 4.90683548158770919 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{x}{{\left(wj \cdot wj\right)}^{3} - 1} \cdot \left(\left(wj - 1\right) \cdot {wj}^{4} + \left(wj - 1\right) \cdot \mathsf{fma}\left(wj, wj, 1\right)\right)}{e^{wj}} + wj\right) - \frac{\mathsf{fma}\left(wj, wj, 1 - wj\right) \cdot wj}{{wj}^{3} + 1}\right) + \mathsf{fma}\left(wj, wj, 1 - wj\right) \cdot \left(\left(-\frac{wj}{{wj}^{3} + 1}\right) + \frac{wj}{{wj}^{3} + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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))))))