VandenBroeck and Keller, Equation (6)

Average Error: 16.7 → 8.8

Time: 9.4s

Precision: 64

• could not determine a ground truth for program body

  1. F = 3.787748117211514e-101
  2. l = -1814629233056857.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.54227830784148189 \cdot 10^{157}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 1.08089408670032465 \cdot 10^{142}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\\ \end{array}\]

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 temp;
	if (((((double) M_PI) * l) <= -1.542278307841482e+157)) {
		temp = ((((double) M_PI) * l) - ((1.0 / F) * ((1.0 / F) * tan(((cbrt((((double) M_PI) * l)) * cbrt((((double) M_PI) * l))) * cbrt((((double) M_PI) * l)))))));
	} else {
		double temp_1;
		if (((((double) M_PI) * l) <= 1.0808940867003247e+142)) {
			temp_1 = ((((double) M_PI) * l) - ((1.0 / F) * (1.0 * (sin((((double) M_PI) * l)) / ((((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)))));
		} else {
			temp_1 = ((((double) M_PI) * l) - ((1.0 / (F * F)) * tan((sqrt(((double) M_PI)) * (sqrt(((double) M_PI)) * l)))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -1.542278307841482e+157

   1. Initial program 20.7

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

   3. Applied *-un-lft-identity 20.7

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

   4. Applied times-frac 20.7

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

   5. Applied associate-*l* 20.7

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

   7. Applied add-cube-cbrt 20.7

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

   if -1.542278307841482e+157 < (* PI l) < 1.0808940867003247e+142

   1. Initial program 15.0

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

   3. Applied *-un-lft-identity 15.0

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

   4. Applied times-frac 15.1

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

   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. Taylor expanded around inf 9.4

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

   7. Taylor expanded around 0 4.2

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

   if 1.0808940867003247e+142 < (* PI l)

   1. Initial program 21.2

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

   3. Applied add-sqr-sqrt 21.2

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

   4. Applied associate-*l* 21.2

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.54227830784148189 \cdot 10^{157}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 1.08089408670032465 \cdot 10^{142}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))