VandenBroeck and Keller, Equation (6)

Average Error: 16.7 → 8.7

Time: 14.4s

Precision: 64

• could not determine a ground truth for program body

  1. F = -0.006342096082774755
  2. l = 2345566735674346.5

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -4.6380634062953026 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 1.8434257583154768 \cdot 10^{144}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\mathsf{fma}\left(\frac{-1}{2}, \left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right), \mathsf{fma}\left(\frac{1}{24}, {\pi}^{4} \cdot {\ell}^{4}, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]

C

double f(double F, double l) {
        double r16442 = atan2(1.0, 0.0);
        double r16443 = l;
        double r16444 = r16442 * r16443;
        double r16445 = 1.0;
        double r16446 = F;
        double r16447 = r16446 * r16446;
        double r16448 = r16445 / r16447;
        double r16449 = tan(r16444);
        double r16450 = r16448 * r16449;
        double r16451 = r16444 - r16450;
        return r16451;
}

↓

double f(double F, double l) {
        double r16452 = atan2(1.0, 0.0);
        double r16453 = l;
        double r16454 = r16452 * r16453;
        double r16455 = -4.6380634062953026e+154;
        bool r16456 = r16454 <= r16455;
        double r16457 = 1.0;
        double r16458 = cbrt(r16457);
        double r16459 = r16458 * r16458;
        double r16460 = F;
        double r16461 = r16459 / r16460;
        double r16462 = sin(r16454);
        double r16463 = r16462 / r16460;
        double r16464 = r16458 * r16463;
        double r16465 = cbrt(r16452);
        double r16466 = r16465 * r16465;
        double r16467 = sqrt(r16452);
        double r16468 = cbrt(r16467);
        double r16469 = r16468 * r16468;
        double r16470 = r16469 * r16453;
        double r16471 = r16466 * r16470;
        double r16472 = cos(r16471);
        double r16473 = r16464 / r16472;
        double r16474 = r16461 * r16473;
        double r16475 = r16454 - r16474;
        double r16476 = 1.8434257583154768e+144;
        bool r16477 = r16454 <= r16476;
        double r16478 = -0.5;
        double r16479 = r16454 * r16454;
        double r16480 = 0.041666666666666664;
        double r16481 = 4.0;
        double r16482 = pow(r16452, r16481);
        double r16483 = pow(r16453, r16481);
        double r16484 = r16482 * r16483;
        double r16485 = 1.0;
        double r16486 = fma(r16480, r16484, r16485);
        double r16487 = fma(r16478, r16479, r16486);
        double r16488 = r16464 / r16487;
        double r16489 = r16461 * r16488;
        double r16490 = r16454 - r16489;
        double r16491 = r16460 * r16460;
        double r16492 = r16457 / r16491;
        double r16493 = cbrt(r16454);
        double r16494 = r16493 * r16493;
        double r16495 = r16494 * r16493;
        double r16496 = tan(r16495);
        double r16497 = r16492 * r16496;
        double r16498 = r16454 - r16497;
        double r16499 = r16477 ? r16490 : r16498;
        double r16500 = r16456 ? r16475 : r16499;
        return r16500;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (* PI l) < -4.6380634062953026e+154

   1. Initial program 21.4

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

   3. Applied add-cube-cbrt 21.4

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

   4. Applied times-frac 21.4

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

   5. Applied associate-*l* 21.4

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

   7. Applied tan-quot 21.4

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

   8. Applied associate-*r/ 21.4

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

   9. Simplified 21.4

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

   11. Applied add-cube-cbrt 21.4

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

   12. Applied associate-*l* 21.4

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

   14. Applied add-sqr-sqrt 21.4

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

   15. Applied cbrt-prod 21.4

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

   if -4.6380634062953026e+154 < (* PI l) < 1.8434257583154768e+144

   1. Initial program 14.8

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

   3. Applied add-cube-cbrt 14.8

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

   4. Applied times-frac 14.8

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

   5. Applied associate-*l* 9.0

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

   7. Applied tan-quot 9.0

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

   8. Applied associate-*r/ 9.0

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

   9. Simplified 9.0

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

   10. Taylor expanded around 0 3.7

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}}\]

   11. Simplified 3.7

       \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right), \mathsf{fma}\left(\frac{1}{24}, {\pi}^{4} \cdot {\ell}^{4}, 1\right)\right)}}\]

   if 1.8434257583154768e+144 < (* PI l)

   1. Initial program 21.6

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

   3. Applied add-cube-cbrt 21.6

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -4.6380634062953026 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 1.8434257583154768 \cdot 10^{144}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\mathsf{fma}\left(\frac{-1}{2}, \left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right), \mathsf{fma}\left(\frac{1}{24}, {\pi}^{4} \cdot {\ell}^{4}, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))