Average Error: 17.1 → 12.7
Time: 28.9s
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 r32172 = atan2(1.0, 0.0);
        double r32173 = l;
        double r32174 = r32172 * r32173;
        double r32175 = 1.0;
        double r32176 = F;
        double r32177 = r32176 * r32176;
        double r32178 = r32175 / r32177;
        double r32179 = tan(r32174);
        double r32180 = r32178 * r32179;
        double r32181 = r32174 - r32180;
        return r32181;
}
↓
double f(double F, double l) {
        double r32182 = atan2(1.0, 0.0);
        double r32183 = l;
        double r32184 = r32182 * r32183;
        double r32185 = 1.0;
        double r32186 = cbrt(r32185);
        double r32187 = r32186 * r32186;
        double r32188 = F;
        double r32189 = r32187 / r32188;
        double r32190 = 1.0;
        double r32191 = cbrt(r32188);
        double r32192 = r32191 * r32191;
        double r32193 = r32190 / r32192;
        double r32194 = r32186 / r32191;
        double r32195 = tan(r32184);
        double r32196 = r32194 * r32195;
        double r32197 = r32193 * r32196;
        double r32198 = r32189 * r32197;
        double r32199 = r32184 - r32198;
        return r32199;
}

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)))))