Average Error: 15.8 → 9.9
Time: 33.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le 89374916.04315051:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\ell \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le 89374916.04315051:\\
\;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\ell \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}{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 r649241 = atan2(1.0, 0.0);
        double r649242 = l;
        double r649243 = r649241 * r649242;
        double r649244 = 1.0;
        double r649245 = F;
        double r649246 = r649245 * r649245;
        double r649247 = r649244 / r649246;
        double r649248 = tan(r649243);
        double r649249 = r649247 * r649248;
        double r649250 = r649243 - r649249;
        return r649250;
}
↓
double f(double F, double l) {
        double r649251 = atan2(1.0, 0.0);
        double r649252 = l;
        double r649253 = r649251 * r649252;
        double r649254 = 89374916.04315051;
        bool r649255 = r649253 <= r649254;
        double r649256 = sqrt(r649251);
        double r649257 = sqrt(r649256);
        double r649258 = r649257 * r649257;
        double r649259 = r649252 * r649258;
        double r649260 = r649258 * r649259;
        double r649261 = tan(r649260);
        double r649262 = F;
        double r649263 = r649261 / r649262;
        double r649264 = r649263 / r649262;
        double r649265 = r649253 - r649264;
        double r649266 = tan(r649253);
        double r649267 = r649266 / r649262;
        double r649268 = /* ERROR: no posit support in C */;
        double r649269 = /* ERROR: no posit support in C */;
        double r649270 = r649269 / r649262;
        double r649271 = r649253 - r649270;
        double r649272 = r649255 ? r649265 : r649271;
        return r649272;
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if (* PI l) < 89374916.04315051

    1. Initial program 13.1

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

    2. Simplified7.6

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

    4. Applied add-sqr-sqrt7.8

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

    5. Applied associate-*l*7.8

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

    7. Applied add-sqr-sqrt7.8

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

    8. Applied sqrt-prod7.7

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

    10. Applied add-sqr-sqrt7.7

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

    11. Applied sqrt-prod7.7

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

    if 89374916.04315051 < (* PI l)

    1. Initial program 23.4

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

    2. Simplified23.4

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

    4. Applied insert-posit1616.3

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le 89374916.04315051:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\ell \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))