Average Error: 17.0 → 12.9
Time: 29.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\tan \left(\sqrt{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}} \cdot \left(\left(\sqrt{\pi} \cdot \ell\right) \cdot \sqrt{\sqrt[3]{\pi}}\right)\right) \cdot \frac{\sqrt{1}}{F}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\tan \left(\sqrt{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}} \cdot \left(\left(\sqrt{\pi} \cdot \ell\right) \cdot \sqrt{\sqrt[3]{\pi}}\right)\right) \cdot \frac{\sqrt{1}}{F}\right)
double f(double F, double l) {
        double r937297 = atan2(1.0, 0.0);
        double r937298 = l;
        double r937299 = r937297 * r937298;
        double r937300 = 1.0;
        double r937301 = F;
        double r937302 = r937301 * r937301;
        double r937303 = r937300 / r937302;
        double r937304 = tan(r937299);
        double r937305 = r937303 * r937304;
        double r937306 = r937299 - r937305;
        return r937306;
}
↓
double f(double F, double l) {
        double r937307 = atan2(1.0, 0.0);
        double r937308 = l;
        double r937309 = r937307 * r937308;
        double r937310 = 1.0;
        double r937311 = sqrt(r937310);
        double r937312 = F;
        double r937313 = r937311 / r937312;
        double r937314 = cbrt(r937307);
        double r937315 = r937314 * r937314;
        double r937316 = sqrt(r937315);
        double r937317 = sqrt(r937307);
        double r937318 = r937317 * r937308;
        double r937319 = sqrt(r937314);
        double r937320 = r937318 * r937319;
        double r937321 = r937316 * r937320;
        double r937322 = tan(r937321);
        double r937323 = r937322 * r937313;
        double r937324 = r937313 * r937323;
        double r937325 = r937309 - r937324;
        return r937325;
}

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 17.0

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

  3. Applied add-sqr-sqrt17.0

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

  4. Applied times-frac17.0

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

  5. Applied associate-*l*12.8

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

  7. Applied add-sqr-sqrt12.9

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

  8. Applied associate-*l*12.9

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

  10. Applied add-cube-cbrt13.0

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

  11. Applied sqrt-prod12.9

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

  12. Applied associate-*l*12.9

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

  13. Final simplification12.9

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))