Average Error: 17.1 → 12.7
Time: 28.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r31335 = atan2(1.0, 0.0);
        double r31336 = l;
        double r31337 = r31335 * r31336;
        double r31338 = 1.0;
        double r31339 = F;
        double r31340 = r31339 * r31339;
        double r31341 = r31338 / r31340;
        double r31342 = tan(r31337);
        double r31343 = r31341 * r31342;
        double r31344 = r31337 - r31343;
        return r31344;
}
↓
double f(double F, double l) {
        double r31345 = atan2(1.0, 0.0);
        double r31346 = l;
        double r31347 = r31345 * r31346;
        double r31348 = 1.0;
        double r31349 = sqrt(r31348);
        double r31350 = F;
        double r31351 = r31349 / r31350;
        double r31352 = 1.0;
        double r31353 = cbrt(r31350);
        double r31354 = r31353 * r31353;
        double r31355 = r31352 / r31354;
        double r31356 = r31349 / r31353;
        double r31357 = tan(r31347);
        double r31358 = r31356 * r31357;
        double r31359 = r31355 * r31358;
        double r31360 = r31351 * r31359;
        double r31361 = r31347 - r31360;
        return r31361;
}

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-sqr-sqrt17.1

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

  4. Applied times-frac17.1

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

  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-cbrt12.7

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

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

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

  9. Applied sqrt-prod12.7

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

  10. Applied times-frac12.7

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

  11. Applied associate-*l*12.7

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

  12. Final simplification12.7

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

Reproduce

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