Average Error: 16.9 → 12.6
Time: 32.3s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\left(\sqrt[3]{\frac{\sqrt{1}}{F}} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\sqrt[3]{F}}\right) \cdot \left(\tan \left(\pi \cdot \ell\right) \cdot \sqrt[3]{\frac{\sqrt{1}}{F}}\right)\right) \cdot \frac{\sqrt{1}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\left(\sqrt[3]{\frac{\sqrt{1}}{F}} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\sqrt[3]{F}}\right) \cdot \left(\tan \left(\pi \cdot \ell\right) \cdot \sqrt[3]{\frac{\sqrt{1}}{F}}\right)\right) \cdot \frac{\sqrt{1}}{F}
double f(double F, double l) {
        double r1027083 = atan2(1.0, 0.0);
        double r1027084 = l;
        double r1027085 = r1027083 * r1027084;
        double r1027086 = 1.0;
        double r1027087 = F;
        double r1027088 = r1027087 * r1027087;
        double r1027089 = r1027086 / r1027088;
        double r1027090 = tan(r1027085);
        double r1027091 = r1027089 * r1027090;
        double r1027092 = r1027085 - r1027091;
        return r1027092;
}
↓
double f(double F, double l) {
        double r1027093 = atan2(1.0, 0.0);
        double r1027094 = l;
        double r1027095 = r1027093 * r1027094;
        double r1027096 = 1.0;
        double r1027097 = sqrt(r1027096);
        double r1027098 = F;
        double r1027099 = r1027097 / r1027098;
        double r1027100 = cbrt(r1027099);
        double r1027101 = cbrt(r1027097);
        double r1027102 = cbrt(r1027098);
        double r1027103 = r1027101 / r1027102;
        double r1027104 = r1027100 * r1027103;
        double r1027105 = tan(r1027095);
        double r1027106 = r1027105 * r1027100;
        double r1027107 = r1027104 * r1027106;
        double r1027108 = r1027107 * r1027099;
        double r1027109 = r1027095 - r1027108;
        return r1027109;
}

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 16.9

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

  3. Applied add-sqr-sqrt16.9

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

  4. Applied times-frac16.9

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

  5. Applied associate-*l*12.5

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

  7. Applied add-cube-cbrt12.6

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

  8. Applied associate-*l*12.6

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

  10. Applied cbrt-div12.6

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

  11. Final simplification12.6

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))