Average Error: 16.7 → 12.7
Time: 23.5s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - 1 \cdot \frac{\frac{\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}}}{\frac{\sqrt[3]{F}}{\tan \left(\pi \cdot \ell\right)}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - 1 \cdot \frac{\frac{\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}}}{\frac{\sqrt[3]{F}}{\tan \left(\pi \cdot \ell\right)}}}{F}
double f(double F, double l) {
        double r28837 = atan2(1.0, 0.0);
        double r28838 = l;
        double r28839 = r28837 * r28838;
        double r28840 = 1.0;
        double r28841 = F;
        double r28842 = r28841 * r28841;
        double r28843 = r28840 / r28842;
        double r28844 = tan(r28839);
        double r28845 = r28843 * r28844;
        double r28846 = r28839 - r28845;
        return r28846;
}
↓
double f(double F, double l) {
        double r28847 = atan2(1.0, 0.0);
        double r28848 = l;
        double r28849 = r28847 * r28848;
        double r28850 = 1.0;
        double r28851 = 1.0;
        double r28852 = F;
        double r28853 = cbrt(r28852);
        double r28854 = r28853 * r28853;
        double r28855 = r28851 / r28854;
        double r28856 = tan(r28849);
        double r28857 = r28853 / r28856;
        double r28858 = r28855 / r28857;
        double r28859 = r28858 / r28852;
        double r28860 = r28850 * r28859;
        double r28861 = r28849 - r28860;
        return r28861;
}

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 div-inv16.7

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

  4. Applied associate-*l*16.7

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

  5. Simplified12.5

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

  7. Applied clear-num12.5

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

  9. Applied *-un-lft-identity12.5

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

  10. Applied add-cube-cbrt12.7

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

  11. Applied times-frac12.7

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

  12. Applied associate-/r*12.7

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

  13. Simplified12.7

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

  14. Final simplification12.7

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

Reproduce

herbie shell --seed 2019305 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))