Average Error: 8.8 → 0.8
Time: 38.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{F}
double f(double F, double l) {
        double r572414 = atan2(1.0, 0.0);
        double r572415 = l;
        double r572416 = r572414 * r572415;
        double r572417 = 1.0;
        double r572418 = F;
        double r572419 = r572418 * r572418;
        double r572420 = r572417 / r572419;
        double r572421 = tan(r572416);
        double r572422 = r572420 * r572421;
        double r572423 = r572416 - r572422;
        return r572423;
}
↓
double f(double F, double l) {
        double r572424 = atan2(1.0, 0.0);
        double r572425 = l;
        double r572426 = r572424 * r572425;
        double r572427 = 1.0;
        double r572428 = F;
        double r572429 = r572427 / r572428;
        double r572430 = tan(r572426);
        double r572431 = r572429 * r572430;
        double r572432 = r572431 * r572429;
        double r572433 = r572426 - r572432;
        return r572433;
}

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 8.8

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

  2. Simplified8.3

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

  4. Applied *-un-lft-identity8.3

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

  5. Applied times-frac0.7

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

  7. Applied div-inv0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))