Average Error: 17.1 → 13.2
Time: 22.7s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt{1}}{F}} \cdot \sqrt[3]{\frac{\sqrt{1}}{F}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt{1}}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \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)
double f(double F, double l) {
        double r31218 = atan2(1.0, 0.0);
        double r31219 = l;
        double r31220 = r31218 * r31219;
        double r31221 = 1.0;
        double r31222 = F;
        double r31223 = r31222 * r31222;
        double r31224 = r31221 / r31223;
        double r31225 = tan(r31220);
        double r31226 = r31224 * r31225;
        double r31227 = r31220 - r31226;
        return r31227;
}
↓
double f(double F, double l) {
        double r31228 = atan2(1.0, 0.0);
        double r31229 = l;
        double r31230 = r31228 * r31229;
        double r31231 = 1.0;
        double r31232 = sqrt(r31231);
        double r31233 = F;
        double r31234 = r31232 / r31233;
        double r31235 = cbrt(r31234);
        double r31236 = r31235 * r31235;
        double r31237 = tan(r31230);
        double r31238 = r31235 * r31237;
        double r31239 = r31236 * r31238;
        double r31240 = r31234 * r31239;
        double r31241 = r31230 - r31240;
        return r31241;
}

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-frac17.2

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

  5. Applied associate-*l*13.1

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

  7. Applied add-cube-cbrt13.2

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

  8. Applied associate-*l*13.2

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

  9. Final simplification13.2

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

Reproduce

herbie shell --seed 2019208 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))