Average Error: 8.5 → 0.6
Time: 36.3s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}
double f(double F, double l) {
        double r678905 = atan2(1.0, 0.0);
        double r678906 = l;
        double r678907 = r678905 * r678906;
        double r678908 = 1.0;
        double r678909 = F;
        double r678910 = r678909 * r678909;
        double r678911 = r678908 / r678910;
        double r678912 = tan(r678907);
        double r678913 = r678911 * r678912;
        double r678914 = r678907 - r678913;
        return r678914;
}
↓
double f(double F, double l) {
        double r678915 = atan2(1.0, 0.0);
        double r678916 = l;
        double r678917 = r678915 * r678916;
        double r678918 = tan(r678917);
        double r678919 = F;
        double r678920 = r678918 / r678919;
        double r678921 = 1.0;
        double r678922 = r678921 / r678919;
        double r678923 = r678920 * r678922;
        double r678924 = r678917 - r678923;
        return r678924;
}

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 8.5

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

  2. Simplified8.1

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

  4. Applied *-un-lft-identity8.1

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

  5. Applied times-frac0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019130 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))