VandenBroeck and Keller, Equation (6)

Average Error: 16.5 → 12.2

Time: 16.7s

Precision: 64

• could not determine a ground truth for program body

  1. F = -8.492309990118079e+19
  2. l = -3479117145685192.5

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

Math

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

↓

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

C

double f(double F, double l) {
        double r17626 = atan2(1.0, 0.0);
        double r17627 = l;
        double r17628 = r17626 * r17627;
        double r17629 = 1.0;
        double r17630 = F;
        double r17631 = r17630 * r17630;
        double r17632 = r17629 / r17631;
        double r17633 = tan(r17628);
        double r17634 = r17632 * r17633;
        double r17635 = r17628 - r17634;
        return r17635;
}

↓

double f(double F, double l) {
        double r17636 = atan2(1.0, 0.0);
        double r17637 = l;
        double r17638 = r17636 * r17637;
        double r17639 = 1.0;
        double r17640 = cbrt(r17639);
        double r17641 = r17640 * r17640;
        double r17642 = F;
        double r17643 = r17641 / r17642;
        double r17644 = r17640 / r17642;
        double r17645 = sqrt(r17636);
        double r17646 = sqrt(r17645);
        double r17647 = r17646 * r17646;
        double r17648 = r17645 * r17637;
        double r17649 = r17647 * r17648;
        double r17650 = tan(r17649);
        double r17651 = r17644 * r17650;
        double r17652 = r17643 * r17651;
        double r17653 = r17638 - r17652;
        return r17653;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Initial program 16.5

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

3. Applied add-cube-cbrt 16.5

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

4. Applied times-frac 16.5

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

5. Applied associate-*l* 12.2

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

7. Applied add-sqr-sqrt 12.3

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

8. Applied associate-*l* 12.3

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

10. Applied add-sqr-sqrt 12.3

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

11. Applied sqrt-prod 12.2

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

12. Final simplification 12.2

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))