Average Error: 16.3 → 9.1
Time: 27.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.1470128979354866 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 6.176150626079544 \cdot 10^{-13}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -1.1470128979354866 \cdot 10^{+18}:\\
\;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\

\mathbf{elif}\;\pi \cdot \ell \le 6.176150626079544 \cdot 10^{-13}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{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 r357443 = atan2(1.0, 0.0);
        double r357444 = l;
        double r357445 = r357443 * r357444;
        double r357446 = 1.0;
        double r357447 = F;
        double r357448 = r357447 * r357447;
        double r357449 = r357446 / r357448;
        double r357450 = tan(r357445);
        double r357451 = r357449 * r357450;
        double r357452 = r357445 - r357451;
        return r357452;
}
↓
double f(double F, double l) {
        double r357453 = atan2(1.0, 0.0);
        double r357454 = l;
        double r357455 = r357453 * r357454;
        double r357456 = -1.1470128979354866e+18;
        bool r357457 = r357455 <= r357456;
        double r357458 = tan(r357455);
        double r357459 = F;
        double r357460 = r357459 * r357459;
        double r357461 = r357458 / r357460;
        double r357462 = /* ERROR: no posit support in C */;
        double r357463 = /* ERROR: no posit support in C */;
        double r357464 = r357455 - r357463;
        double r357465 = 6.176150626079544e-13;
        bool r357466 = r357455 <= r357465;
        double r357467 = 1.0;
        double r357468 = r357467 / r357459;
        double r357469 = r357458 / r357459;
        double r357470 = r357468 * r357469;
        double r357471 = r357455 - r357470;
        double r357472 = /* ERROR: no posit support in C */;
        double r357473 = /* ERROR: no posit support in C */;
        double r357474 = r357473 / r357459;
        double r357475 = r357455 - r357474;
        double r357476 = r357466 ? r357471 : r357475;
        double r357477 = r357457 ? r357464 : r357476;
        return r357477;
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -1.1470128979354866e+18

    1. Initial program 23.3

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

    2. Simplified23.3

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

    4. Applied insert-posit1615.5

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

    if -1.1470128979354866e+18 < (* PI l) < 6.176150626079544e-13

    1. Initial program 9.1

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

    2. Simplified8.6

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

    4. Applied *-un-lft-identity8.6

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

    5. Applied times-frac0.8

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

    if 6.176150626079544e-13 < (* PI l)

    1. Initial program 22.0

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

    2. Simplified22.0

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

    4. Applied associate-/r*22.0

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

    6. Applied insert-posit1617.2

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.1470128979354866 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 6.176150626079544 \cdot 10^{-13}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))