Average Error: 16.9 → 12.6
Time: 26.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\left(\frac{\sqrt[3]{\sqrt{1}}}{\sqrt[3]{F}} \cdot \sqrt[3]{\frac{\sqrt{1}}{F}}\right) \cdot \left(\frac{\sqrt[3]{\sqrt{1}}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right) \cdot \frac{\sqrt{1}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\left(\frac{\sqrt[3]{\sqrt{1}}}{\sqrt[3]{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) \cdot \frac{\sqrt{1}}{F}
double f(double F, double l) {
        double r859297 = atan2(1.0, 0.0);
        double r859298 = l;
        double r859299 = r859297 * r859298;
        double r859300 = 1.0;
        double r859301 = F;
        double r859302 = r859301 * r859301;
        double r859303 = r859300 / r859302;
        double r859304 = tan(r859299);
        double r859305 = r859303 * r859304;
        double r859306 = r859299 - r859305;
        return r859306;
}
↓
double f(double F, double l) {
        double r859307 = atan2(1.0, 0.0);
        double r859308 = l;
        double r859309 = r859307 * r859308;
        double r859310 = 1.0;
        double r859311 = sqrt(r859310);
        double r859312 = cbrt(r859311);
        double r859313 = F;
        double r859314 = cbrt(r859313);
        double r859315 = r859312 / r859314;
        double r859316 = r859311 / r859313;
        double r859317 = cbrt(r859316);
        double r859318 = r859315 * r859317;
        double r859319 = tan(r859309);
        double r859320 = r859315 * r859319;
        double r859321 = r859318 * r859320;
        double r859322 = r859321 * r859316;
        double r859323 = r859309 - r859322;
        return r859323;
}

Error

Bits error versus F

Bits error versus l

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.9

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

  3. Applied add-sqr-sqrt16.9

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

  4. Applied times-frac16.9

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

  5. Applied associate-*l*12.5

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

  7. Applied add-cube-cbrt12.6

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

  8. Applied associate-*l*12.6

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

  10. Applied cbrt-div12.6

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

  12. Applied cbrt-div12.6

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

  13. Final simplification12.6

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

Reproduce

herbie shell --seed 2019171 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))