Average Error: 16.9 → 12.9
Time: 26.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)
double f(double F, double l) {
        double r26025 = atan2(1.0, 0.0);
        double r26026 = l;
        double r26027 = r26025 * r26026;
        double r26028 = 1.0;
        double r26029 = F;
        double r26030 = r26029 * r26029;
        double r26031 = r26028 / r26030;
        double r26032 = tan(r26027);
        double r26033 = r26031 * r26032;
        double r26034 = r26027 - r26033;
        return r26034;
}
↓
double f(double F, double l) {
        double r26035 = atan2(1.0, 0.0);
        double r26036 = l;
        double r26037 = r26035 * r26036;
        double r26038 = 1.0;
        double r26039 = F;
        double r26040 = r26038 / r26039;
        double r26041 = cbrt(r26040);
        double r26042 = r26041 * r26041;
        double r26043 = 1.0;
        double r26044 = tan(r26037);
        double r26045 = r26043 * r26044;
        double r26046 = r26045 / r26039;
        double r26047 = r26041 * r26046;
        double r26048 = r26042 * r26047;
        double r26049 = r26037 - r26048;
        return r26049;
}

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 *-un-lft-identity16.9

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

  4. Applied times-frac16.9

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

  5. Applied associate-*l*12.7

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

  7. Applied associate-*l/12.7

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

  9. Applied add-cube-cbrt12.9

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

  10. Applied associate-*l*12.9

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

  11. Final simplification12.9

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

Reproduce

herbie shell --seed 2019325 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))