Average Error: 17.1 → 12.7
Time: 33.7s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - 1 \cdot \frac{\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - 1 \cdot \frac{\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}}}{F}
double f(double F, double l) {
        double r25315 = atan2(1.0, 0.0);
        double r25316 = l;
        double r25317 = r25315 * r25316;
        double r25318 = 1.0;
        double r25319 = F;
        double r25320 = r25319 * r25319;
        double r25321 = r25318 / r25320;
        double r25322 = tan(r25317);
        double r25323 = r25321 * r25322;
        double r25324 = r25317 - r25323;
        return r25324;
}
↓
double f(double F, double l) {
        double r25325 = atan2(1.0, 0.0);
        double r25326 = l;
        double r25327 = r25325 * r25326;
        double r25328 = 1.0;
        double r25329 = 1.0;
        double r25330 = F;
        double r25331 = cbrt(r25330);
        double r25332 = r25331 * r25331;
        double r25333 = r25329 / r25332;
        double r25334 = tan(r25327);
        double r25335 = r25334 / r25331;
        double r25336 = r25333 * r25335;
        double r25337 = r25336 / r25330;
        double r25338 = r25328 * r25337;
        double r25339 = r25327 - r25338;
        return r25339;
}

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 div-inv17.1

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

  4. Applied associate-*l*17.1

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

  5. Simplified12.5

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

  7. Applied add-cube-cbrt12.7

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

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

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

  9. Applied times-frac12.7

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

  10. Final simplification12.7

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

Reproduce

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