Average Error: 8.5 → 0.7
Time: 37.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}
double f(double F, double l) {
        double r555618 = atan2(1.0, 0.0);
        double r555619 = l;
        double r555620 = r555618 * r555619;
        double r555621 = 1.0;
        double r555622 = F;
        double r555623 = r555622 * r555622;
        double r555624 = r555621 / r555623;
        double r555625 = tan(r555620);
        double r555626 = r555624 * r555625;
        double r555627 = r555620 - r555626;
        return r555627;
}
↓
double f(double F, double l) {
        double r555628 = atan2(1.0, 0.0);
        double r555629 = l;
        double r555630 = r555628 * r555629;
        double r555631 = tan(r555630);
        double r555632 = F;
        double r555633 = r555631 / r555632;
        double r555634 = r555633 / r555632;
        double r555635 = r555630 - r555634;
        return r555635;
}

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. Simplified8.0

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

  4. Applied associate-/r*0.7

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

  6. Applied add-sqr-sqrt1.0

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

  7. Applied associate-*l*1.0

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

  8. Taylor expanded around -inf 0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))