Average Error: 16.8 → 8.9
Time: 31.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.9407008719341217 \cdot 10^{+25}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 1.897007709166529 \cdot 10^{+17}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -2.9407008719341217 \cdot 10^{+25}:\\
\;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\

\mathbf{elif}\;\pi \cdot \ell \le 1.897007709166529 \cdot 10^{+17}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\

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

\end{array}
double f(double F, double l) {
        double r763940 = atan2(1.0, 0.0);
        double r763941 = l;
        double r763942 = r763940 * r763941;
        double r763943 = 1.0;
        double r763944 = F;
        double r763945 = r763944 * r763944;
        double r763946 = r763943 / r763945;
        double r763947 = tan(r763942);
        double r763948 = r763946 * r763947;
        double r763949 = r763942 - r763948;
        return r763949;
}
↓
double f(double F, double l) {
        double r763950 = atan2(1.0, 0.0);
        double r763951 = l;
        double r763952 = r763950 * r763951;
        double r763953 = -2.9407008719341217e+25;
        bool r763954 = r763952 <= r763953;
        double r763955 = tan(r763952);
        double r763956 = F;
        double r763957 = r763956 * r763956;
        double r763958 = r763955 / r763957;
        double r763959 = /* ERROR: no posit support in C */;
        double r763960 = /* ERROR: no posit support in C */;
        double r763961 = r763952 - r763960;
        double r763962 = 1.897007709166529e+17;
        bool r763963 = r763952 <= r763962;
        double r763964 = 1.0;
        double r763965 = r763964 / r763956;
        double r763966 = r763956 / r763955;
        double r763967 = r763964 / r763966;
        double r763968 = r763965 * r763967;
        double r763969 = r763952 - r763968;
        double r763970 = r763963 ? r763969 : r763961;
        double r763971 = r763954 ? r763961 : r763970;
        return r763971;
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if (* PI l) < -2.9407008719341217e+25 or 1.897007709166529e+17 < (* PI l)

    1. Initial program 24.1

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

    2. Simplified24.1

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

    4. Applied insert-posit1616.4

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

    if -2.9407008719341217e+25 < (* PI l) < 1.897007709166529e+17

    1. Initial program 9.6

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

    2. Simplified9.1

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

    4. Applied *-un-lft-identity9.1

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

    5. Applied times-frac1.3

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

    7. Applied clear-num1.3

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \color{blue}{\frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.9407008719341217 \cdot 10^{+25}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 1.897007709166529 \cdot 10^{+17}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))