Average Error: 17.5 → 8.8
Time: 8.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -7.92727466488188767 \cdot 10^{162}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 3.309098575533877 \cdot 10^{144}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right) \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right) \cdot \tan \left(\pi \cdot \ell\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -7.92727466488188767 \cdot 10^{162}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)\\

\mathbf{elif}\;\pi \cdot \ell \le 3.309098575533877 \cdot 10^{144}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\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) \cdot F}\right)\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \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)\\

\end{array}
double f(double F, double l) {
        double r15170 = atan2(1.0, 0.0);
        double r15171 = l;
        double r15172 = r15170 * r15171;
        double r15173 = 1.0;
        double r15174 = F;
        double r15175 = r15174 * r15174;
        double r15176 = r15173 / r15175;
        double r15177 = tan(r15172);
        double r15178 = r15176 * r15177;
        double r15179 = r15172 - r15178;
        return r15179;
}
↓
double f(double F, double l) {
        double r15180 = atan2(1.0, 0.0);
        double r15181 = l;
        double r15182 = r15180 * r15181;
        double r15183 = -7.927274664881888e+162;
        bool r15184 = r15182 <= r15183;
        double r15185 = 1.0;
        double r15186 = F;
        double r15187 = r15185 / r15186;
        double r15188 = 1.0;
        double r15189 = r15188 / r15186;
        double r15190 = cbrt(r15180);
        double r15191 = r15190 * r15190;
        double r15192 = r15190 * r15181;
        double r15193 = r15191 * r15192;
        double r15194 = tan(r15193);
        double r15195 = r15189 * r15194;
        double r15196 = r15187 * r15195;
        double r15197 = r15182 - r15196;
        double r15198 = 3.309098575533877e+144;
        bool r15199 = r15182 <= r15198;
        double r15200 = sin(r15182);
        double r15201 = 0.041666666666666664;
        double r15202 = 4.0;
        double r15203 = pow(r15180, r15202);
        double r15204 = pow(r15181, r15202);
        double r15205 = r15203 * r15204;
        double r15206 = r15201 * r15205;
        double r15207 = r15206 + r15185;
        double r15208 = 0.5;
        double r15209 = 2.0;
        double r15210 = pow(r15180, r15209);
        double r15211 = pow(r15181, r15209);
        double r15212 = r15210 * r15211;
        double r15213 = r15208 * r15212;
        double r15214 = r15207 - r15213;
        double r15215 = r15214 * r15186;
        double r15216 = r15200 / r15215;
        double r15217 = r15188 * r15216;
        double r15218 = r15187 * r15217;
        double r15219 = r15182 - r15218;
        double r15220 = r15186 * r15186;
        double r15221 = r15188 / r15220;
        double r15222 = cbrt(r15221);
        double r15223 = r15222 * r15222;
        double r15224 = r15223 * r15222;
        double r15225 = tan(r15182);
        double r15226 = r15224 * r15225;
        double r15227 = r15182 - r15226;
        double r15228 = r15199 ? r15219 : r15227;
        double r15229 = r15184 ? r15197 : r15228;
        return r15229;
}

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) < -7.927274664881888e+162

    1. Initial program 19.5

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

    3. Applied *-un-lft-identity19.5

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

    4. Applied times-frac19.5

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

    5. Applied associate-*l*19.5

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

    7. Applied add-cube-cbrt19.4

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

    8. Applied associate-*l*19.4

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

    if -7.927274664881888e+162 < (* PI l) < 3.309098575533877e+144

    1. Initial program 16.5

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

    3. Applied *-un-lft-identity16.5

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

    4. Applied times-frac16.5

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

    5. Applied associate-*l*10.4

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

    6. Taylor expanded around inf 10.3

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

    7. Taylor expanded around 0 4.6

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

    if 3.309098575533877e+144 < (* PI l)

    1. Initial program 20.8

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

    3. Applied add-cube-cbrt20.8

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -7.92727466488188767 \cdot 10^{162}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 3.309098575533877 \cdot 10^{144}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\sqrt[3]{\frac{1}{F \cdot F}} \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right) \cdot \sqrt[3]{\frac{1}{F \cdot F}}\right) \cdot \tan \left(\pi \cdot \ell\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))