Average Error: 16.7 → 12.7
Time: 29.7s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r28651 = atan2(1.0, 0.0);
        double r28652 = l;
        double r28653 = r28651 * r28652;
        double r28654 = 1.0;
        double r28655 = F;
        double r28656 = r28655 * r28655;
        double r28657 = r28654 / r28656;
        double r28658 = tan(r28653);
        double r28659 = r28657 * r28658;
        double r28660 = r28653 - r28659;
        return r28660;
}
↓
double f(double F, double l) {
        double r28661 = atan2(1.0, 0.0);
        double r28662 = l;
        double r28663 = r28661 * r28662;
        double r28664 = 1.0;
        double r28665 = F;
        double r28666 = r28664 / r28665;
        double r28667 = 1.0;
        double r28668 = r28667 / r28665;
        double r28669 = cbrt(r28668);
        double r28670 = r28669 * r28669;
        double r28671 = tan(r28663);
        double r28672 = r28669 * r28671;
        double r28673 = r28670 * r28672;
        double r28674 = r28666 * r28673;
        double r28675 = r28663 - r28674;
        return r28675;
}

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*12.6

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

  7. Applied add-cube-cbrt12.7

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

  8. Applied associate-*l*12.7

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

  9. Final simplification12.7

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))