VandenBroeck and Keller, Equation (6)

Average Error: 16.4 → 8.0

Time: 7.6s

Precision: 64

• could not determine a ground truth for program body

  1. F = 3.831489363817351e+222
  2. l = 4243037263318092.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -4.88463047305792789 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 9.0915089882537537 \cdot 10^{150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{\left|F\right|}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot \left|F\right|}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - e^{\log \left(\frac{1}{F \cdot F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\\ \end{array}\]

C

double code(double F, double l) {
	return ((double) (((double) (((double) M_PI) * l)) - ((double) (((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)) <= -4.884630473057928e+154)) {
		VAR = ((double) (((double) (((double) M_PI) * l)) - ((double) (((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))))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) M_PI) * l)) <= 9.091508988253754e+150)) {
			VAR_1 = ((double) (((double) (((double) M_PI) * l)) - ((double) (((double) (((double) (1.0 * ((double) sin(((double) (((double) M_PI) * l)))))) / ((double) fabs(F)))) / ((double) (((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)))))))) * ((double) fabs(F))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) M_PI) * l)) - ((double) (((double) exp(((double) log(((double) (1.0 / ((double) (F * F)))))))) * ((double) tan(((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) < -4.884630473057928e+154

   1. Initial program 19.6

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

   3. Applied add-cube-cbrt 19.6

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

   if -4.884630473057928e+154 < (* PI l) < 9.091508988253754e+150

   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 tan-quot 15.3

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

   4. Applied add-sqr-sqrt 15.3

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

   5. Applied associate-/r* 15.3

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

   6. Applied frac-times 14.8

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

   7. Simplified 14.8

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

   8. Simplified 9.0

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

   9. Taylor expanded around 0 3.7

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

   if 9.091508988253754e+150 < (* PI l)

   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-exp-log 41.2

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

   4. Applied add-exp-log 41.2

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

   5. Applied prod-exp 41.2

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

   6. Applied add-exp-log 41.2

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

   7. Applied div-exp 41.2

      \[\leadsto \pi \cdot \ell - \color{blue}{e^{\log 1 - \left(\log F + \log F\right)}} \cdot \tan \left(\pi \cdot \ell\right)\]

   8. Simplified 19.3

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

4. Final simplification 8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -4.88463047305792789 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 9.0915089882537537 \cdot 10^{150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{\left|F\right|}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot \left|F\right|}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - e^{\log \left(\frac{1}{F \cdot F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))