Average Error: 16.9 → 12.9
Time: 26.7s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \left(1 \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \left(1 \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)
double f(double F, double l) {
        double r26261 = atan2(1.0, 0.0);
        double r26262 = l;
        double r26263 = r26261 * r26262;
        double r26264 = 1.0;
        double r26265 = F;
        double r26266 = r26265 * r26265;
        double r26267 = r26264 / r26266;
        double r26268 = tan(r26263);
        double r26269 = r26267 * r26268;
        double r26270 = r26263 - r26269;
        return r26270;
}
↓
double f(double F, double l) {
        double r26271 = atan2(1.0, 0.0);
        double r26272 = l;
        double r26273 = r26271 * r26272;
        double r26274 = 1.0;
        double r26275 = F;
        double r26276 = r26274 / r26275;
        double r26277 = cbrt(r26276);
        double r26278 = r26277 * r26277;
        double r26279 = 1.0;
        double r26280 = tan(r26273);
        double r26281 = r26280 / r26275;
        double r26282 = r26279 * r26281;
        double r26283 = r26277 * r26282;
        double r26284 = r26278 * r26283;
        double r26285 = r26273 - r26284;
        return r26285;
}

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 16.9

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

  3. Applied *-un-lft-identity16.9

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

  4. Applied times-frac16.9

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

  5. Applied associate-*l*12.7

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

  7. Applied div-inv12.7

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

  8. Applied associate-*l*12.7

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

  9. Simplified12.7

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

  11. Applied add-cube-cbrt12.9

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

  12. Applied associate-*l*12.9

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

  13. Final simplification12.9

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))