VandenBroeck and Keller, Equation (6)

Average Error: 16.4 → 9.0

Time: 9.3s

Precision: binary64

• could not determine a ground truth for program body

  1. F = -1.8087614978070984e+288
  2. l = 3003819648806307.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -6.20801884469397047 \cdot 10^{155}:\\ \;\;\;\;\pi \cdot \ell - 1 \cdot \frac{\left(\frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{\sqrt[3]{\cos \left(\pi \cdot \ell\right)}} \cdot \frac{\frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{\sqrt[3]{\cos \left(\pi \cdot \ell\right)}}}{\sqrt[3]{F} \cdot \sqrt[3]{F}}\right) \cdot \frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{F}}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le 1.4066591555946186 \cdot 10^{126}:\\ \;\;\;\;\pi \cdot \ell - 1 \cdot \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{1 + \left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + \pi \cdot \left(\frac{-1}{2} \cdot \left(\pi \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - 1 \cdot \frac{\frac{\tan \left(\sqrt{\pi \cdot \ell} \cdot \sqrt{\pi \cdot \ell}\right)}{F}}{F}\\ \end{array}\]

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)) <= -6.20801884469397e+155)) {
		VAR = ((double) (((double) (((double) M_PI) * l)) - ((double) (1.0 * ((double) (((double) (((double) (((double) (((double) cbrt(((double) sin(((double) (((double) M_PI) * l)))))) / ((double) cbrt(((double) cos(((double) (((double) M_PI) * l)))))))) * ((double) (((double) (((double) cbrt(((double) sin(((double) (((double) M_PI) * l)))))) / ((double) cbrt(((double) cos(((double) (((double) M_PI) * l)))))))) / ((double) (((double) cbrt(F)) * ((double) cbrt(F)))))))) * ((double) (((double) cbrt(((double) sin(((double) (((double) M_PI) * l)))))) / ((double) (((double) cbrt(((double) cos(((double) (((double) M_PI) * l)))))) * ((double) cbrt(F)))))))) / F))))));
	} else {
		double VAR_1;
		if ((((double) (((double) M_PI) * l)) <= 1.4066591555946186e+126)) {
			VAR_1 = ((double) (((double) (((double) M_PI) * l)) - ((double) (1.0 * ((double) (((double) (((double) (((double) sin(((double) (((double) M_PI) * l)))) / ((double) (1.0 + ((double) (((double) (0.041666666666666664 * ((double) (((double) pow(((double) M_PI), 4.0)) * ((double) pow(l, 4.0)))))) + ((double) (((double) M_PI) * ((double) (-0.5 * ((double) (((double) M_PI) * ((double) (l * l)))))))))))))) / F)) / F))))));
		} else {
			VAR_1 = ((double) (((double) (((double) M_PI) * l)) - ((double) (1.0 * ((double) (((double) (((double) tan(((double) (((double) sqrt(((double) (((double) M_PI) * l)))) * ((double) sqrt(((double) (((double) M_PI) * l)))))))) / F)) / F))))));
		}
		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) < -6.20801884469397047e155

   1. Initial program 19.9

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

   2. Simplified 19.9

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

   4. Applied associate-/r* 19.9

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

   6. Applied tan-quot 19.9

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

   8. Applied add-cube-cbrt 19.9

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

   9. Applied add-cube-cbrt 19.9

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

   10. Applied add-cube-cbrt 19.9

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

   11. Applied times-frac 19.9

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

   12. Applied times-frac 19.9

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

   13. Simplified 19.9

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

   14. Simplified 19.9

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

   if -6.20801884469397047e155 < (* PI l) < 1.4066591555946186e126

   1. Initial program 14.8

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

   2. Simplified 14.4

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

   4. Applied associate-/r* 9.1

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

   6. Applied tan-quot 9.1

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

   7. Taylor expanded around 0 4.3

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

   8. Simplified 4.3

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

   if 1.4066591555946186e126 < (* PI l)

   1. Initial program 20.3

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

   2. Simplified 20.3

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

   4. Applied associate-/r* 20.3

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

   6. Applied add-sqr-sqrt 20.3

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -6.20801884469397047 \cdot 10^{155}:\\ \;\;\;\;\pi \cdot \ell - 1 \cdot \frac{\left(\frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{\sqrt[3]{\cos \left(\pi \cdot \ell\right)}} \cdot \frac{\frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{\sqrt[3]{\cos \left(\pi \cdot \ell\right)}}}{\sqrt[3]{F} \cdot \sqrt[3]{F}}\right) \cdot \frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{F}}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le 1.4066591555946186 \cdot 10^{126}:\\ \;\;\;\;\pi \cdot \ell - 1 \cdot \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{1 + \left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + \pi \cdot \left(\frac{-1}{2} \cdot \left(\pi \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - 1 \cdot \frac{\frac{\tan \left(\sqrt{\pi \cdot \ell} \cdot \sqrt{\pi \cdot \ell}\right)}{F}}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))