Average Error: 16.1 → 12.2
Time: 8.4s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\left({\left(\sqrt[3]{\sqrt[3]{\pi}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\left({\left(\sqrt[3]{\sqrt[3]{\pi}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r15193 = atan2(1.0, 0.0);
        double r15194 = l;
        double r15195 = r15193 * r15194;
        double r15196 = 1.0;
        double r15197 = F;
        double r15198 = r15197 * r15197;
        double r15199 = r15196 / r15198;
        double r15200 = tan(r15195);
        double r15201 = r15199 * r15200;
        double r15202 = r15195 - r15201;
        return r15202;
}
↓
double f(double F, double l) {
        double r15203 = atan2(1.0, 0.0);
        double r15204 = l;
        double r15205 = r15203 * r15204;
        double r15206 = 1.0;
        double r15207 = cbrt(r15206);
        double r15208 = r15207 * r15207;
        double r15209 = F;
        double r15210 = r15208 / r15209;
        double r15211 = r15207 / r15209;
        double r15212 = cbrt(r15203);
        double r15213 = cbrt(r15212);
        double r15214 = 5.0;
        double r15215 = pow(r15213, r15214);
        double r15216 = r15215 * r15213;
        double r15217 = r15212 * r15204;
        double r15218 = r15216 * r15217;
        double r15219 = tan(r15218);
        double r15220 = r15211 * r15219;
        double r15221 = r15210 * r15220;
        double r15222 = r15205 - r15221;
        return r15222;
}

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. Initial program 16.1

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

  3. Applied add-cube-cbrt16.1

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

  4. Applied times-frac16.1

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

  5. Applied associate-*l*12.2

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

  7. Applied add-cube-cbrt12.4

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

  8. Applied associate-*l*12.4

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

  10. Applied add-cube-cbrt12.2

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

  11. Applied associate-*r*12.2

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

  12. Simplified12.2

    \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\pi}}\right)}^{5}} \cdot \sqrt[3]{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)\]

  13. Final simplification12.2

    \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\left({\left(\sqrt[3]{\sqrt[3]{\pi}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)\]

Reproduce

herbie shell --seed 2020018 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))