Average Error: 16.8 → 13.0
Time: 9.9s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{\left(1 \cdot \sqrt[3]{F}\right) \cdot \sqrt[3]{F}} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{\left(1 \cdot \sqrt[3]{F}\right) \cdot \sqrt[3]{F}} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}}\right)
double f(double F, double l) {
        double r20258 = atan2(1.0, 0.0);
        double r20259 = l;
        double r20260 = r20258 * r20259;
        double r20261 = 1.0;
        double r20262 = F;
        double r20263 = r20262 * r20262;
        double r20264 = r20261 / r20263;
        double r20265 = tan(r20260);
        double r20266 = r20264 * r20265;
        double r20267 = r20260 - r20266;
        return r20267;
}
↓
double f(double F, double l) {
        double r20268 = atan2(1.0, 0.0);
        double r20269 = l;
        double r20270 = r20268 * r20269;
        double r20271 = 1.0;
        double r20272 = sqrt(r20271);
        double r20273 = F;
        double r20274 = r20272 / r20273;
        double r20275 = 1.0;
        double r20276 = cbrt(r20273);
        double r20277 = r20275 * r20276;
        double r20278 = r20277 * r20276;
        double r20279 = r20272 / r20278;
        double r20280 = tan(r20270);
        double r20281 = r20280 / r20276;
        double r20282 = r20279 * r20281;
        double r20283 = r20274 * r20282;
        double r20284 = r20270 - r20283;
        return r20284;
}

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 16.8

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

  3. Applied add-sqr-sqrt16.8

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

  4. Applied times-frac16.8

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

  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 associate-*l/12.8

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

  9. Applied add-cube-cbrt13.0

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

  10. Applied times-frac13.0

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

  12. Applied *-un-lft-identity13.0

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

  13. Final simplification13.0

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

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))