Average Error: 16.7 → 12.4
Time: 9.0s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r16833 = atan2(1.0, 0.0);
        double r16834 = l;
        double r16835 = r16833 * r16834;
        double r16836 = 1.0;
        double r16837 = F;
        double r16838 = r16837 * r16837;
        double r16839 = r16836 / r16838;
        double r16840 = tan(r16835);
        double r16841 = r16839 * r16840;
        double r16842 = r16835 - r16841;
        return r16842;
}
↓
double f(double F, double l) {
        double r16843 = atan2(1.0, 0.0);
        double r16844 = l;
        double r16845 = r16843 * r16844;
        double r16846 = 1.0;
        double r16847 = F;
        double r16848 = r16846 / r16847;
        double r16849 = 1.0;
        double r16850 = r16849 / r16847;
        double r16851 = sqrt(r16843);
        double r16852 = r16851 * r16844;
        double r16853 = r16851 * r16852;
        double r16854 = tan(r16853);
        double r16855 = r16850 * r16854;
        double r16856 = r16848 * r16855;
        double r16857 = r16845 - r16856;
        return r16857;
}

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-frac16.8

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

  5. Applied associate-*l*12.3

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

  7. Applied add-sqr-sqrt12.4

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

  8. Applied associate-*l*12.4

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

  9. Final simplification12.4

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

Reproduce

herbie shell --seed 2020027 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))