Average Error: 8.8 → 0.9
Time: 39.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\tan \left(\left(\sqrt{\pi} \cdot \ell\right) \cdot \sqrt{\pi}\right) \cdot \frac{1}{F}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\tan \left(\left(\sqrt{\pi} \cdot \ell\right) \cdot \sqrt{\pi}\right) \cdot \frac{1}{F}\right)
double f(double F, double l) {
        double r528605 = atan2(1.0, 0.0);
        double r528606 = l;
        double r528607 = r528605 * r528606;
        double r528608 = 1.0;
        double r528609 = F;
        double r528610 = r528609 * r528609;
        double r528611 = r528608 / r528610;
        double r528612 = tan(r528607);
        double r528613 = r528611 * r528612;
        double r528614 = r528607 - r528613;
        return r528614;
}
↓
double f(double F, double l) {
        double r528615 = atan2(1.0, 0.0);
        double r528616 = l;
        double r528617 = r528615 * r528616;
        double r528618 = 1.0;
        double r528619 = F;
        double r528620 = r528618 / r528619;
        double r528621 = sqrt(r528615);
        double r528622 = r528621 * r528616;
        double r528623 = r528622 * r528621;
        double r528624 = tan(r528623);
        double r528625 = r528624 * r528620;
        double r528626 = r528620 * r528625;
        double r528627 = r528617 - r528626;
        return r528627;
}

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 8.8

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

  2. Simplified8.3

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

  4. Applied *-un-lft-identity8.3

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

  5. Applied times-frac0.7

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

  7. Applied div-inv0.8

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

  9. Applied add-sqr-sqrt1.0

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

  10. Applied associate-*l*0.9

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

  11. Final simplification0.9

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

Reproduce

herbie shell --seed 2019119 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))