Average Error: 16.7 → 12.4
Time: 30.0s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \left(\left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right) \cdot \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{\sqrt[3]{F}}\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \left(\left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right) \cdot \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{\sqrt[3]{F}}\right)\right)
double f(double F, double l) {
        double r25490 = atan2(1.0, 0.0);
        double r25491 = l;
        double r25492 = r25490 * r25491;
        double r25493 = 1.0;
        double r25494 = F;
        double r25495 = r25494 * r25494;
        double r25496 = r25493 / r25495;
        double r25497 = tan(r25492);
        double r25498 = r25496 * r25497;
        double r25499 = r25492 - r25498;
        return r25499;
}
↓
double f(double F, double l) {
        double r25500 = atan2(1.0, 0.0);
        double r25501 = l;
        double r25502 = r25500 * r25501;
        double r25503 = 1.0;
        double r25504 = F;
        double r25505 = r25503 / r25504;
        double r25506 = 1.0;
        double r25507 = tan(r25502);
        double r25508 = r25507 / r25504;
        double r25509 = cbrt(r25508);
        double r25510 = r25509 * r25509;
        double r25511 = cbrt(r25507);
        double r25512 = cbrt(r25504);
        double r25513 = r25511 / r25512;
        double r25514 = r25510 * r25513;
        double r25515 = r25506 * r25514;
        double r25516 = r25505 * r25515;
        double r25517 = r25502 - r25516;
        return r25517;
}

Error

Bits error versus F

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.7

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

  3. Applied *-un-lft-identity16.7

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

  4. Applied times-frac16.7

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

  5. Applied associate-*l*12.2

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

  7. Applied div-inv12.2

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

  8. Applied associate-*l*12.2

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

  9. Simplified12.2

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

  11. Applied add-cube-cbrt12.4

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

  13. Applied cbrt-div12.4

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

  14. Final simplification12.4

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

Reproduce

herbie shell --seed 2019326 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))