Average Error: 16.6 → 10.6
Time: 34.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le 43788683678766.93:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\sqrt{\pi} \cdot \ell\right)\right) \cdot \frac{1}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le 43788683678766.93:\\
\;\;\;\;\pi \cdot \ell - \frac{\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\sqrt{\pi} \cdot \ell\right)\right) \cdot \frac{1}{F}}{F}\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\

\end{array}
double f(double F, double l) {
        double r707682 = atan2(1.0, 0.0);
        double r707683 = l;
        double r707684 = r707682 * r707683;
        double r707685 = 1.0;
        double r707686 = F;
        double r707687 = r707686 * r707686;
        double r707688 = r707685 / r707687;
        double r707689 = tan(r707684);
        double r707690 = r707688 * r707689;
        double r707691 = r707684 - r707690;
        return r707691;
}
↓
double f(double F, double l) {
        double r707692 = atan2(1.0, 0.0);
        double r707693 = l;
        double r707694 = r707692 * r707693;
        double r707695 = 43788683678766.93;
        bool r707696 = r707694 <= r707695;
        double r707697 = sqrt(r707692);
        double r707698 = sqrt(r707697);
        double r707699 = r707698 * r707698;
        double r707700 = r707697 * r707693;
        double r707701 = r707699 * r707700;
        double r707702 = tan(r707701);
        double r707703 = 1.0;
        double r707704 = F;
        double r707705 = r707703 / r707704;
        double r707706 = r707702 * r707705;
        double r707707 = r707706 / r707704;
        double r707708 = r707694 - r707707;
        double r707709 = tan(r707694);
        double r707710 = r707709 / r707704;
        double r707711 = /* ERROR: no posit support in C */;
        double r707712 = /* ERROR: no posit support in C */;
        double r707713 = r707712 / r707704;
        double r707714 = r707694 - r707713;
        double r707715 = r707696 ? r707708 : r707714;
        return r707715;
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if (* PI l) < 43788683678766.93

    1. Initial program 14.5

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

    2. Simplified8.9

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

    4. Applied div-inv9.0

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

    6. Applied add-sqr-sqrt9.1

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

    7. Applied associate-*l*9.1

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

    9. Applied add-sqr-sqrt9.0

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

    if 43788683678766.93 < (* PI l)

    1. Initial program 22.7

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

    2. Simplified22.7

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

    4. Applied div-inv22.7

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

    6. Applied insert-posit1615.5

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

    7. Simplified15.5

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}}{F}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le 43788683678766.93:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\sqrt{\pi} \cdot \ell\right)\right) \cdot \frac{1}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))