VandenBroeck and Keller, Equation (6)

Average Error: 16.6 → 8.5

Time: 10.0s

Precision: binary64

• could not determine a ground truth for program body

  1. F = -1.678408177611356e-233
  2. l = 2340279284988967.5

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.16227614669724016 \cdot 10^{160}:\\ \;\;\;\;\pi \cdot \ell - \frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot {F}^{2}}\\ \mathbf{elif}\;\pi \cdot \ell \le 5.20170593347389519 \cdot 10^{146}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]

C

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

↓

double code(double F, double l) {
	double VAR;
	if ((((double) (((double) M_PI) * l)) <= -2.1622761466972402e+160)) {
		VAR = ((double) (((double) (((double) M_PI) * l)) - (((double) (1.0 * ((double) sin(((double) (((double) M_PI) * l)))))) / ((double) (((double) cos(((double) (((double) M_PI) * l)))) * ((double) pow(F, 2.0)))))));
	} else {
		double VAR_1;
		if ((((double) (((double) M_PI) * l)) <= 5.201705933473895e+146)) {
			VAR_1 = ((double) (((double) (((double) M_PI) * l)) - ((double) ((1.0 / F) * (((double) (1.0 * (((double) sin(((double) (((double) M_PI) * l)))) / ((double) (((double) (((double) (0.041666666666666664 * ((double) (((double) pow(((double) M_PI), 4.0)) * ((double) pow(l, 4.0)))))) + 1.0)) - ((double) (0.5 * ((double) (((double) pow(((double) M_PI), 2.0)) * ((double) pow(l, 2.0))))))))))) / F)))));
		} else {
			VAR_1 = ((double) (((double) (((double) M_PI) * l)) - ((double) ((1.0 / ((double) (F * F))) * ((double) tan(((double) (((double) (((double) cbrt(((double) (((double) M_PI) * l)))) * ((double) cbrt(((double) (((double) M_PI) * l)))))) * ((double) cbrt(((double) (((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) < -2.16227614669724016e160

   1. Initial program 19.4

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

   3. Applied tan-quot 19.4

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

   4. Applied frac-times 19.4

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

   5. Simplified 19.4

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

   if -2.16227614669724016e160 < (* PI l) < 5.20170593347389519e146

   1. Initial program 15.2

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

   3. Applied *-un-lft-identity 15.2

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

   4. Applied times-frac 15.2

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

   5. Applied associate-*l* 9.8

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

   6. Simplified 9.8

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

   8. Applied tan-quot 9.8

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

   9. Taylor expanded around 0 4.1

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

   if 5.20170593347389519e146 < (* PI l)

   1. Initial program 20.9

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

   3. Applied add-cube-cbrt 20.9

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.16227614669724016 \cdot 10^{160}:\\ \;\;\;\;\pi \cdot \ell - \frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot {F}^{2}}\\ \mathbf{elif}\;\pi \cdot \ell \le 5.20170593347389519 \cdot 10^{146}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))