Average Error: 8.5 → 0.7
Time: 41.6s
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 r453421 = atan2(1.0, 0.0);
        double r453422 = l;
        double r453423 = r453421 * r453422;
        double r453424 = 1.0;
        double r453425 = F;
        double r453426 = r453425 * r453425;
        double r453427 = r453424 / r453426;
        double r453428 = tan(r453423);
        double r453429 = r453427 * r453428;
        double r453430 = r453423 - r453429;
        return r453430;
}
↓
double f(double F, double l) {
        double r453431 = atan2(1.0, 0.0);
        double r453432 = l;
        double r453433 = r453431 * r453432;
        double r453434 = 1.0;
        double r453435 = F;
        double r453436 = r453434 / r453435;
        double r453437 = tan(r453433);
        double r453438 = r453434 / r453437;
        double r453439 = r453436 / r453438;
        double r453440 = r453439 / r453435;
        double r453441 = r453433 - r453440;
        return r453441;
}

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 8.5

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

  2. Simplified0.7

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

  4. Applied *-un-lft-identity0.7

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

  5. Applied associate-/l*0.7

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

  7. Applied div-inv0.7

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

  8. Applied associate-/r*0.7

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

  9. 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 2019142 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))