Jmat.Real.lambertw, newton loop step

Average Error: 14.1 → 0.9

Time: 5.9s

Precision: 64

• could not determine a ground truth for program body

  1. wj = 1.611838088146848e+293
  2. x = -1.5548261699289447e+154

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le 6.66261844019268921 \cdot 10^{-9}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) - \left(\sqrt[3]{2} \cdot \left(wj \cdot x\right)\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)\right) + \left(\sqrt[3]{2} \cdot \left(wj \cdot x\right)\right) \cdot \left(\left(-\sqrt[3]{2} \cdot \sqrt[3]{2}\right) + \sqrt[3]{2} \cdot \sqrt[3]{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \mathsf{fma}\left(\frac{x}{{wj}^{3} + 1}, \frac{wj \cdot wj + \left(1 - wj\right)}{e^{wj}}, wj\right), -\left(wj - 1\right) \cdot \frac{wj}{wj \cdot wj - 1}\right) + \mathsf{fma}\left(-\left(wj - 1\right), \frac{wj}{wj \cdot wj - 1}, \left(wj - 1\right) \cdot \frac{wj}{wj \cdot wj - 1}\right)\\ \end{array}\]

C

double code(double wj, double x) {
	return ((double) (wj - ((double) (((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 <= 6.662618440192689e-09)) {
		VAR = ((double) (((double) (((double) fma(wj, wj, x)) - ((double) (((double) (((double) cbrt(2.0)) * ((double) (wj * x)))) * ((double) (((double) cbrt(2.0)) * ((double) cbrt(2.0)))))))) + ((double) (((double) (((double) cbrt(2.0)) * ((double) (wj * x)))) * ((double) (((double) -(((double) (((double) cbrt(2.0)) * ((double) cbrt(2.0)))))) + ((double) (((double) cbrt(2.0)) * ((double) cbrt(2.0))))))))));
	} else {
		VAR = ((double) (((double) fma(1.0, ((double) fma(((double) (x / ((double) (((double) pow(wj, 3.0)) + 1.0)))), ((double) (((double) (((double) (wj * wj)) + ((double) (1.0 - wj)))) / ((double) exp(wj)))), wj)), ((double) -(((double) (((double) (wj - 1.0)) * ((double) (wj / ((double) (((double) (wj * wj)) - 1.0)))))))))) + ((double) fma(((double) -(((double) (wj - 1.0)))), ((double) (wj / ((double) (((double) (wj * wj)) - 1.0)))), ((double) (((double) (wj - 1.0)) * ((double) (wj / ((double) (((double) (wj * wj)) - 1.0))))))))));
	}
	return VAR;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	14.1
Target	13.6
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 < 6.662618440192689e-09

   1. Initial program 13.9

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

   2. Simplified 13.9

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

   3. Taylor expanded around 0 0.9

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

   5. Applied add-cube-cbrt 0.9

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

   6. Applied associate-*l* 0.9

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

   7. Applied add-sqr-sqrt 29.1

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

   8. Applied prod-diff 29.1

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

   9. Simplified 0.9

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

   10. Simplified 0.9

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

   if 6.662618440192689e-09 < wj

   1. Initial program 20.4

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

   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 *-un-lft-identity 3.3

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

   5. Applied flip3-+ 3.3

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

   6. Applied associate-/r/ 3.3

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

   7. Applied times-frac 3.3

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

   8. Simplified 3.3

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

   9. Simplified 3.3

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

   11. Applied flip-+ 3.4

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

   12. Applied associate-/r/ 3.4

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

   13. Applied add-sqr-sqrt 12.7

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

   14. Applied prod-diff 12.6

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

   15. Simplified 3.2

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

   16. Simplified 3.2

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 6.66261844019268921 \cdot 10^{-9}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) - \left(\sqrt[3]{2} \cdot \left(wj \cdot x\right)\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)\right) + \left(\sqrt[3]{2} \cdot \left(wj \cdot x\right)\right) \cdot \left(\left(-\sqrt[3]{2} \cdot \sqrt[3]{2}\right) + \sqrt[3]{2} \cdot \sqrt[3]{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \mathsf{fma}\left(\frac{x}{{wj}^{3} + 1}, \frac{wj \cdot wj + \left(1 - wj\right)}{e^{wj}}, wj\right), -\left(wj - 1\right) \cdot \frac{wj}{wj \cdot wj - 1}\right) + \mathsf{fma}\left(-\left(wj - 1\right), \frac{wj}{wj \cdot wj - 1}, \left(wj - 1\right) \cdot \frac{wj}{wj \cdot wj - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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))))))