Average Error: 16.9 → 12.5
Time: 29.9s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \ell\right)\right)\right)\]
\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(\sqrt{\pi} \cdot \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r25055 = atan2(1.0, 0.0);
        double r25056 = l;
        double r25057 = r25055 * r25056;
        double r25058 = 1.0;
        double r25059 = F;
        double r25060 = r25059 * r25059;
        double r25061 = r25058 / r25060;
        double r25062 = tan(r25057);
        double r25063 = r25061 * r25062;
        double r25064 = r25057 - r25063;
        return r25064;
}
↓
double f(double F, double l) {
        double r25065 = atan2(1.0, 0.0);
        double r25066 = l;
        double r25067 = r25065 * r25066;
        double r25068 = 1.0;
        double r25069 = cbrt(r25068);
        double r25070 = r25069 * r25069;
        double r25071 = F;
        double r25072 = r25070 / r25071;
        double r25073 = r25069 / r25071;
        double r25074 = sqrt(r25065);
        double r25075 = sqrt(r25074);
        double r25076 = r25075 * r25075;
        double r25077 = r25076 * r25066;
        double r25078 = r25074 * r25077;
        double r25079 = tan(r25078);
        double r25080 = r25073 * r25079;
        double r25081 = r25072 * r25080;
        double r25082 = r25067 - r25081;
        return r25082;
}

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-cube-cbrt16.9

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

  4. Applied times-frac16.9

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

  5. Applied associate-*l*12.5

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

  7. Applied add-sqr-sqrt12.6

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

  8. Applied associate-*l*12.6

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

  10. Applied add-sqr-sqrt12.6

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

  11. Applied sqrt-prod12.5

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

  12. Final simplification12.5

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

Reproduce

herbie shell --seed 2019303 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))