VandenBroeck and Keller, Equation (6)

Average Error: 16.6 → 8.3

Time: 8.2s

Precision: 64

• could not determine a ground truth for program body

  1. F = 5.1117133960315295e-239
  2. l = -3057773833859993.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.0845581642214246 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 7.7154867931676 \cdot 10^{137}:\\ \;\;\;\;\pi \cdot \ell - {\left(\frac{\frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}{F}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)\\ \end{array}\]

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) <= -2.0845581642214246e+154)) {
		VAR = ((((double) M_PI) * l) - ((1.0 / (F * F)) * tan(((cbrt((((double) M_PI) * l)) * cbrt((((double) M_PI) * l))) * cbrt((((double) M_PI) * l))))));
	} else {
		double VAR_1;
		if (((((double) M_PI) * l) <= 7.715486793167634e+137)) {
			VAR_1 = ((((double) M_PI) * l) - pow((((1.0 * sin((((double) M_PI) * l))) / (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))))))) / F), 1.0));
		} else {
			VAR_1 = ((((double) M_PI) * l) - ((1.0 / (F * F)) * tan(((((double) M_PI) * (cbrt(l) * cbrt(l))) * cbrt(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) < -2.0845581642214246e+154

   1. Initial program 19.0

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

   3. Applied add-cube-cbrt 19.0

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

   if -2.0845581642214246e+154 < (* PI l) < 7.715486793167634e+137

   1. Initial program 15.3

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

   3. Applied *-un-lft-identity 15.3

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

   4. Applied times-frac 15.3

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

   5. Applied associate-*l* 9.4

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

   7. Applied tan-quot 9.4

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

   8. Applied frac-times 9.4

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

   10. Applied pow1 9.4

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

   11. Applied pow1 9.4

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

   12. Applied pow-prod-down 9.4

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

   13. Simplified 9.4

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

   14. Taylor expanded around 0 3.7

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

   15. Simplified 3.7

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

   if 7.715486793167634e+137 < (* 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

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

   4. Applied associate-*r* 20.5

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

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.0845581642214246 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 7.7154867931676 \cdot 10^{137}:\\ \;\;\;\;\pi \cdot \ell - {\left(\frac{\frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {\pi}^{4}, {\ell}^{4}, 1 - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}{F}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))