Average Error: 17.1 → 12.7
Time: 29.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{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 r32173 = atan2(1.0, 0.0);
        double r32174 = l;
        double r32175 = r32173 * r32174;
        double r32176 = 1.0;
        double r32177 = F;
        double r32178 = r32177 * r32177;
        double r32179 = r32176 / r32178;
        double r32180 = tan(r32175);
        double r32181 = r32179 * r32180;
        double r32182 = r32175 - r32181;
        return r32182;
}
↓
double f(double F, double l) {
        double r32183 = atan2(1.0, 0.0);
        double r32184 = l;
        double r32185 = r32183 * r32184;
        double r32186 = 1.0;
        double r32187 = cbrt(r32186);
        double r32188 = r32187 * r32187;
        double r32189 = F;
        double r32190 = r32188 / r32189;
        double r32191 = 1.0;
        double r32192 = cbrt(r32189);
        double r32193 = r32192 * r32192;
        double r32194 = r32191 / r32193;
        double r32195 = r32187 / r32192;
        double r32196 = tan(r32185);
        double r32197 = r32195 * r32196;
        double r32198 = r32194 * r32197;
        double r32199 = r32190 * r32198;
        double r32200 = r32185 - r32199;
        return r32200;
}

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 17.1

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

  3. Applied add-cube-cbrt17.1

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

  4. Applied times-frac17.1

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

  5. Applied associate-*l*12.5

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

  7. Applied add-cube-cbrt12.7

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

  8. Applied *-un-lft-identity12.7

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

  9. Applied cbrt-prod12.7

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

  10. Applied times-frac12.7

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

  11. Applied associate-*l*12.7

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

  12. Final simplification12.7

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))