Average Error: 16.4 → 9.2
Time: 9.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -5.229943255210993702040420524981676341617 \cdot 10^{158}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 1.570116288117577749110006291473035572901 \cdot 10^{117}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{1}{\frac{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -5.229943255210993702040420524981676341617 \cdot 10^{158}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\\

\mathbf{elif}\;\pi \cdot \ell \le 1.570116288117577749110006291473035572901 \cdot 10^{117}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{1}{\frac{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}}\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\

\end{array}
double f(double F, double l) {
        double r16793 = atan2(1.0, 0.0);
        double r16794 = l;
        double r16795 = r16793 * r16794;
        double r16796 = 1.0;
        double r16797 = F;
        double r16798 = r16797 * r16797;
        double r16799 = r16796 / r16798;
        double r16800 = tan(r16795);
        double r16801 = r16799 * r16800;
        double r16802 = r16795 - r16801;
        return r16802;
}
↓
double f(double F, double l) {
        double r16803 = atan2(1.0, 0.0);
        double r16804 = l;
        double r16805 = r16803 * r16804;
        double r16806 = -5.229943255210994e+158;
        bool r16807 = r16805 <= r16806;
        double r16808 = 1.0;
        double r16809 = sqrt(r16808);
        double r16810 = F;
        double r16811 = r16809 / r16810;
        double r16812 = sin(r16805);
        double r16813 = r16809 * r16812;
        double r16814 = sqrt(r16803);
        double r16815 = r16814 * r16804;
        double r16816 = r16814 * r16815;
        double r16817 = cos(r16816);
        double r16818 = r16810 * r16817;
        double r16819 = r16813 / r16818;
        double r16820 = r16811 * r16819;
        double r16821 = r16805 - r16820;
        double r16822 = 1.5701162881175777e+117;
        bool r16823 = r16805 <= r16822;
        double r16824 = 1.0;
        double r16825 = 0.041666666666666664;
        double r16826 = 4.0;
        double r16827 = pow(r16803, r16826);
        double r16828 = pow(r16804, r16826);
        double r16829 = r16827 * r16828;
        double r16830 = r16825 * r16829;
        double r16831 = r16830 + r16824;
        double r16832 = 0.5;
        double r16833 = 2.0;
        double r16834 = pow(r16803, r16833);
        double r16835 = pow(r16804, r16833);
        double r16836 = r16834 * r16835;
        double r16837 = r16832 * r16836;
        double r16838 = r16831 - r16837;
        double r16839 = r16810 * r16838;
        double r16840 = r16839 / r16813;
        double r16841 = r16824 / r16840;
        double r16842 = r16811 * r16841;
        double r16843 = r16805 - r16842;
        double r16844 = r16810 * r16810;
        double r16845 = r16808 / r16844;
        double r16846 = cbrt(r16805);
        double r16847 = r16846 * r16846;
        double r16848 = r16847 * r16846;
        double r16849 = tan(r16848);
        double r16850 = r16845 * r16849;
        double r16851 = r16805 - r16850;
        double r16852 = r16823 ? r16843 : r16851;
        double r16853 = r16807 ? r16821 : r16852;
        return r16853;
}

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. Split input into 3 regimes

  2. if (* PI l) < -5.229943255210994e+158

    1. Initial program 20.5

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

    3. Applied add-sqr-sqrt20.5

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

    4. Applied times-frac20.5

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

    5. Applied associate-*l*20.5

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

    7. Applied tan-quot20.5

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

    8. Applied frac-times20.5

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

    10. Applied add-sqr-sqrt20.6

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

    11. Applied associate-*l*20.6

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

    if -5.229943255210994e+158 < (* PI l) < 1.5701162881175777e+117

    1. Initial program 14.3

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

    3. Applied add-sqr-sqrt14.3

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

    4. Applied times-frac14.4

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

    5. Applied associate-*l*8.6

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

    7. Applied tan-quot8.6

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

    8. Applied frac-times8.6

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

    10. Applied clear-num8.6

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

    11. Taylor expanded around 0 4.1

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{1}{\frac{F \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}}\]

    if 1.5701162881175777e+117 < (* PI l)

    1. Initial program 21.6

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

    3. Applied add-cube-cbrt21.6

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -5.229943255210993702040420524981676341617 \cdot 10^{158}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 1.570116288117577749110006291473035572901 \cdot 10^{117}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{1}{\frac{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))