Average Error: 17.1 → 12.7
Time: 28.3s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r30521 = atan2(1.0, 0.0);
        double r30522 = l;
        double r30523 = r30521 * r30522;
        double r30524 = 1.0;
        double r30525 = F;
        double r30526 = r30525 * r30525;
        double r30527 = r30524 / r30526;
        double r30528 = tan(r30523);
        double r30529 = r30527 * r30528;
        double r30530 = r30523 - r30529;
        return r30530;
}
↓
double f(double F, double l) {
        double r30531 = atan2(1.0, 0.0);
        double r30532 = l;
        double r30533 = r30531 * r30532;
        double r30534 = 1.0;
        double r30535 = F;
        double r30536 = r30534 / r30535;
        double r30537 = 1.0;
        double r30538 = sqrt(r30537);
        double r30539 = cbrt(r30535);
        double r30540 = r30539 * r30539;
        double r30541 = r30538 / r30540;
        double r30542 = r30538 / r30539;
        double r30543 = tan(r30533);
        double r30544 = r30542 * r30543;
        double r30545 = r30541 * r30544;
        double r30546 = r30536 * r30545;
        double r30547 = r30533 - r30546;
        return r30547;
}

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 17.1

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

  3. Applied *-un-lft-identity17.1

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

  4. Applied times-frac17.1

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

  5. Applied associate-*l*12.5

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

  7. Applied add-cube-cbrt12.7

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

  8. Applied add-sqr-sqrt12.7

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

  9. Applied times-frac12.7

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

  10. Applied associate-*l*12.7

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

  11. Final simplification12.7

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

Reproduce

herbie shell --seed 2019323 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))