Average Error: 16.5 → 12.6
Time: 9.3s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r16949 = atan2(1.0, 0.0);
        double r16950 = l;
        double r16951 = r16949 * r16950;
        double r16952 = 1.0;
        double r16953 = F;
        double r16954 = r16953 * r16953;
        double r16955 = r16952 / r16954;
        double r16956 = tan(r16951);
        double r16957 = r16955 * r16956;
        double r16958 = r16951 - r16957;
        return r16958;
}
↓
double f(double F, double l) {
        double r16959 = atan2(1.0, 0.0);
        double r16960 = l;
        double r16961 = r16959 * r16960;
        double r16962 = 1.0;
        double r16963 = F;
        double r16964 = r16962 / r16963;
        double r16965 = 1.0;
        double r16966 = cbrt(r16965);
        double r16967 = r16966 * r16966;
        double r16968 = cbrt(r16963);
        double r16969 = r16968 * r16968;
        double r16970 = r16967 / r16969;
        double r16971 = r16966 / r16968;
        double r16972 = tan(r16961);
        double r16973 = r16971 * r16972;
        double r16974 = r16970 * r16973;
        double r16975 = r16964 * r16974;
        double r16976 = r16961 - r16975;
        return r16976;
}

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 16.5

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

  3. Applied *-un-lft-identity16.5

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

  4. Applied times-frac16.5

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

  5. Applied associate-*l*12.5

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

  7. Applied add-cube-cbrt12.6

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

  8. Applied add-cube-cbrt12.6

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

  9. Applied times-frac12.6

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

  10. Applied associate-*l*12.6

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

  11. Final simplification12.6

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

Reproduce

herbie shell --seed 2020020 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))