VandenBroeck and Keller, Equation (6)

Average Error: 16.4 → 8.5

Time: 8.2s

Precision: 64

• could not determine a ground truth for program body

  1. F = -3.904036790610166e+41
  2. l = 1225813426851089.5

Math

\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.47850802229549789 \cdot 10^{162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\left(\left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{1}}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 2.3173396121667722 \cdot 10^{147}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \ell\right)\right)\right)\\ \end{array}\]

C

double code(double F, double l) {
	return ((((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l))));
}

↓

double code(double F, double l) {
	double VAR;
	if (((((double) M_PI) * l) <= -1.4785080222954979e+162)) {
		VAR = ((((double) M_PI) * l) - (((cbrt(1.0) * cbrt(1.0)) / F) * (((((cbrt(sin((((double) M_PI) * l))) * cbrt(sin((((double) M_PI) * l)))) * cbrt(sin((((double) M_PI) * l)))) * cbrt(1.0)) / F) / cos((sqrt(((double) M_PI)) * (sqrt(((double) M_PI)) * l))))));
	} else {
		double VAR_1;
		if (((((double) M_PI) * l) <= 2.3173396121667722e+147)) {
			VAR_1 = ((((double) M_PI) * l) - (((cbrt(1.0) * cbrt(1.0)) / F) * (((sin((((double) M_PI) * l)) * cbrt(1.0)) / F) / fma((0.041666666666666664 * pow(((double) M_PI), 4.0)), pow(l, 4.0), (1.0 - (0.5 * (pow(((double) M_PI), 2.0) * pow(l, 2.0))))))));
		} else {
			VAR_1 = ((((double) M_PI) * l) - (((cbrt(1.0) * cbrt(1.0)) / F) * ((cbrt(1.0) / F) * tan((sqrt(((double) M_PI)) * ((sqrt(sqrt(((double) M_PI))) * sqrt(sqrt(((double) M_PI)))) * l))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -1.4785080222954979e+162

   1. Initial program 19.3

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 19.3

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

   4. Applied times-frac 19.3

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

   5. Applied associate-*l* 19.3

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 19.2

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot \ell\right)\right)\]

   8. Applied associate-*l* 19.3

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \color{blue}{\left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\right)\]
   9. Using strategy rm

   10. Applied tan-quot 19.3

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \color{blue}{\frac{\sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}}\right)\]

   11. Applied associate-*r/ 19.3

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \color{blue}{\frac{\frac{\sqrt[3]{1}}{F} \cdot \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}}\]

   12. Simplified 19.3

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 19.3

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\color{blue}{\left(\left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right)} \cdot \sqrt[3]{1}}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\]

   if -1.4785080222954979e+162 < (* PI l) < 2.3173396121667722e+147

   1. Initial program 15.1

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 15.1

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

   4. Applied times-frac 15.2

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

   5. Applied associate-*l* 9.4

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 9.5

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot \ell\right)\right)\]

   8. Applied associate-*l* 9.5

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \color{blue}{\left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\right)\]
   9. Using strategy rm

   10. Applied tan-quot 9.5

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \color{blue}{\frac{\sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}}\right)\]

   11. Applied associate-*r/ 9.5

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \color{blue}{\frac{\frac{\sqrt[3]{1}}{F} \cdot \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}}\]

   12. Simplified 9.4

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\]

   13. Taylor expanded around 0 4.3

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\color{blue}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}}\]

   14. Simplified 4.3

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}\]

   if 2.3173396121667722e+147 < (* PI l)

   1. Initial program 20.5

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 20.5

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

   4. Applied times-frac 20.5

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

   5. Applied associate-*l* 20.5

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 20.5

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot \ell\right)\right)\]

   8. Applied associate-*l* 20.5

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \color{blue}{\left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\right)\]
   9. Using strategy rm

   10. Applied add-sqr-sqrt 20.5

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}} \cdot \ell\right)\right)\right)\]

   11. Applied sqrt-prod 20.5

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\color{blue}{\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)} \cdot \ell\right)\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.47850802229549789 \cdot 10^{162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\left(\left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{1}}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 2.3173396121667722 \cdot 10^{147}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \ell\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))