Average Error: 8.6 → 0.6
Time: 30.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{\cos \left(\pi \cdot \ell\right)}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{\cos \left(\pi \cdot \ell\right)}}{F}
double f(double F, double l) {
        double r282623 = atan2(1.0, 0.0);
        double r282624 = l;
        double r282625 = r282623 * r282624;
        double r282626 = 1.0;
        double r282627 = F;
        double r282628 = r282627 * r282627;
        double r282629 = r282626 / r282628;
        double r282630 = tan(r282625);
        double r282631 = r282629 * r282630;
        double r282632 = r282625 - r282631;
        return r282632;
}
↓
double f(double F, double l) {
        double r282633 = atan2(1.0, 0.0);
        double r282634 = l;
        double r282635 = r282633 * r282634;
        double r282636 = sin(r282635);
        double r282637 = F;
        double r282638 = r282636 / r282637;
        double r282639 = cos(r282635);
        double r282640 = r282638 / r282639;
        double r282641 = r282640 / r282637;
        double r282642 = r282635 - r282641;
        return r282642;
}

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 8.6

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

  2. Simplified8.1

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

  3. Taylor expanded around inf 8.1

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

  4. Simplified0.6

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

  6. Applied add-sqr-sqrt0.9

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

  7. Applied associate-*l*0.9

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

  8. Taylor expanded around -inf 0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))