Average Error: 8.0 → 0.7
Time: 37.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\frac{1}{F}}{\frac{1}{\tan \left(\pi \cdot \ell\right)}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{\frac{1}{F}}{\frac{1}{\tan \left(\pi \cdot \ell\right)}}}{F}
double f(double F, double l) {
        double r455265 = atan2(1.0, 0.0);
        double r455266 = l;
        double r455267 = r455265 * r455266;
        double r455268 = 1.0;
        double r455269 = F;
        double r455270 = r455269 * r455269;
        double r455271 = r455268 / r455270;
        double r455272 = tan(r455267);
        double r455273 = r455271 * r455272;
        double r455274 = r455267 - r455273;
        return r455274;
}
↓
double f(double F, double l) {
        double r455275 = atan2(1.0, 0.0);
        double r455276 = l;
        double r455277 = r455275 * r455276;
        double r455278 = 1.0;
        double r455279 = F;
        double r455280 = r455278 / r455279;
        double r455281 = tan(r455277);
        double r455282 = r455278 / r455281;
        double r455283 = r455280 / r455282;
        double r455284 = r455283 / r455279;
        double r455285 = r455277 - r455284;
        return r455285;
}

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 8.0

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

  2. Simplified7.6

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

  4. Applied associate-/r*0.6

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

  6. Applied clear-num0.6

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

  8. Applied div-inv0.7

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

  9. Applied associate-/r*0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))