Average Error: 16.3 → 8.1
Time: 23.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(F \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{3}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(F \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{3}}}{F}
double f(double F, double l) {
        double r397702 = atan2(1.0, 0.0);
        double r397703 = l;
        double r397704 = r397702 * r397703;
        double r397705 = 1.0;
        double r397706 = F;
        double r397707 = r397706 * r397706;
        double r397708 = r397705 / r397707;
        double r397709 = tan(r397704);
        double r397710 = r397708 * r397709;
        double r397711 = r397704 - r397710;
        return r397711;
}
↓
double f(double F, double l) {
        double r397712 = atan2(1.0, 0.0);
        double r397713 = l;
        double r397714 = r397712 * r397713;
        double r397715 = 1.0;
        double r397716 = F;
        double r397717 = r397716 / r397714;
        double r397718 = r397716 * r397714;
        double r397719 = 0.3333333333333333;
        double r397720 = r397718 * r397719;
        double r397721 = r397717 - r397720;
        double r397722 = r397715 / r397721;
        double r397723 = r397722 / r397716;
        double r397724 = r397714 - r397723;
        return r397724;
}

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 16.3

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

  2. Simplified16.0

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

  4. Applied associate-/r*12.1

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

  6. Applied clear-num12.1

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

  7. Taylor expanded around 0 8.1

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

  8. Final simplification8.1

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

Reproduce

herbie shell --seed 2019156 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))