Average Error: 16.3 → 8.4
Time: 16.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -5.274938856688908 \cdot 10^{164} \lor \neg \left(\pi \cdot \ell \le 2.5077607211161675 \cdot 10^{153}\right):\\ \;\;\;\;\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -5.274938856688908 \cdot 10^{164} \lor \neg \left(\pi \cdot \ell \le 2.5077607211161675 \cdot 10^{153}\right):\\
\;\;\;\;\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\\

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

\end{array}
double f(double F, double l) {
        double r65 = atan2(1.0, 0.0);
        double r66 = l;
        double r67 = r65 * r66;
        double r68 = 1.0;
        double r69 = F;
        double r70 = r69 * r69;
        double r71 = r68 / r70;
        double r72 = tan(r67);
        double r73 = r71 * r72;
        double r74 = r67 - r73;
        return r74;
}
↓
double f(double F, double l) {
        double r75 = atan2(1.0, 0.0);
        double r76 = l;
        double r77 = r75 * r76;
        double r78 = -5.274938856688908e+164;
        bool r79 = r77 <= r78;
        double r80 = 2.5077607211161675e+153;
        bool r81 = r77 <= r80;
        double r82 = !r81;
        bool r83 = r79 || r82;
        double r84 = 1.0;
        double r85 = F;
        double r86 = r85 * r85;
        double r87 = r84 / r86;
        double r88 = cbrt(r87);
        double r89 = r88 * r88;
        double r90 = tan(r77);
        double r91 = r88 * r90;
        double r92 = r89 * r91;
        double r93 = r77 - r92;
        double r94 = cbrt(r84);
        double r95 = r94 * r94;
        double r96 = r95 / r85;
        double r97 = sin(r77);
        double r98 = r94 * r97;
        double r99 = 0.041666666666666664;
        double r100 = 4.0;
        double r101 = pow(r75, r100);
        double r102 = pow(r76, r100);
        double r103 = r101 * r102;
        double r104 = r99 * r103;
        double r105 = 1.0;
        double r106 = r104 + r105;
        double r107 = 0.5;
        double r108 = 2.0;
        double r109 = pow(r75, r108);
        double r110 = pow(r76, r108);
        double r111 = r109 * r110;
        double r112 = r107 * r111;
        double r113 = r106 - r112;
        double r114 = r85 * r113;
        double r115 = r98 / r114;
        double r116 = r96 * r115;
        double r117 = r77 - r116;
        double r118 = r83 ? r93 : r117;
        return r118;
}

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 2 regimes

  2. if (* PI l) < -5.274938856688908e+164 or 2.5077607211161675e+153 < (* PI l)

    1. Initial program 19.2

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

    3. Applied add-cube-cbrt19.2

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right) \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

    4. Applied associate-*l*19.2

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]

    if -5.274938856688908e+164 < (* PI l) < 2.5077607211161675e+153

    1. Initial program 15.3

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

    3. Applied add-cube-cbrt15.3

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

    4. Applied times-frac15.3

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

    5. Applied associate-*l*9.8

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
    6. Using strategy rm

    7. Applied tan-quot9.8

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}\right)\]

    8. Applied frac-times9.7

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \color{blue}{\frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}}\]

    9. Taylor expanded around 0 4.5

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -5.274938856688908 \cdot 10^{164} \lor \neg \left(\pi \cdot \ell \le 2.5077607211161675 \cdot 10^{153}\right):\\ \;\;\;\;\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))