Average Error: 16.8 → 12.7
Time: 30.5s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt{1}}{F}}\right) \cdot \left(\frac{\sqrt[3]{\sqrt{1}}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt{1}}{F}}\right) \cdot \left(\frac{\sqrt[3]{\sqrt{1}}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r37132 = atan2(1.0, 0.0);
        double r37133 = l;
        double r37134 = r37132 * r37133;
        double r37135 = 1.0;
        double r37136 = F;
        double r37137 = r37136 * r37136;
        double r37138 = r37135 / r37137;
        double r37139 = tan(r37134);
        double r37140 = r37138 * r37139;
        double r37141 = r37134 - r37140;
        return r37141;
}
↓
double f(double F, double l) {
        double r37142 = atan2(1.0, 0.0);
        double r37143 = l;
        double r37144 = r37142 * r37143;
        double r37145 = 1.0;
        double r37146 = sqrt(r37145);
        double r37147 = F;
        double r37148 = r37146 / r37147;
        double r37149 = cbrt(r37148);
        double r37150 = r37149 * r37149;
        double r37151 = cbrt(r37146);
        double r37152 = cbrt(r37147);
        double r37153 = r37151 / r37152;
        double r37154 = tan(r37144);
        double r37155 = r37153 * r37154;
        double r37156 = r37150 * r37155;
        double r37157 = r37148 * r37156;
        double r37158 = r37144 - r37157;
        return r37158;
}

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 16.8

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

  3. Applied add-sqr-sqrt16.8

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

  4. Applied times-frac16.8

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

  5. Applied associate-*l*12.6

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

  7. Applied add-cube-cbrt12.7

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

  8. Applied associate-*l*12.7

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

  10. Applied cbrt-div12.7

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

  11. Final simplification12.7

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))