Average Error: 9.0 → 1.2
Time: 41.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}}\right)
double f(double F, double l) {
        double r624049 = atan2(1.0, 0.0);
        double r624050 = l;
        double r624051 = r624049 * r624050;
        double r624052 = 1.0;
        double r624053 = F;
        double r624054 = r624053 * r624053;
        double r624055 = r624052 / r624054;
        double r624056 = tan(r624051);
        double r624057 = r624055 * r624056;
        double r624058 = r624051 - r624057;
        return r624058;
}
↓
double f(double F, double l) {
        double r624059 = atan2(1.0, 0.0);
        double r624060 = l;
        double r624061 = r624059 * r624060;
        double r624062 = 1.0;
        double r624063 = F;
        double r624064 = r624062 / r624063;
        double r624065 = cbrt(r624063);
        double r624066 = r624065 * r624065;
        double r624067 = r624062 / r624066;
        double r624068 = tan(r624061);
        double r624069 = r624068 / r624065;
        double r624070 = r624067 * r624069;
        double r624071 = r624064 * r624070;
        double r624072 = r624061 - r624071;
        return r624072;
}

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 9.0

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

  2. Simplified8.5

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

  4. Applied *-un-lft-identity8.5

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

  5. Applied times-frac0.8

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

  7. Applied add-cube-cbrt1.2

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

  8. Applied *-un-lft-identity1.2

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

  9. Applied times-frac1.2

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

  10. Final simplification1.2

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))