Average Error: 8.4 → 0.7
Time: 44.6s
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 r741101 = atan2(1.0, 0.0);
        double r741102 = l;
        double r741103 = r741101 * r741102;
        double r741104 = 1.0;
        double r741105 = F;
        double r741106 = r741105 * r741105;
        double r741107 = r741104 / r741106;
        double r741108 = tan(r741103);
        double r741109 = r741107 * r741108;
        double r741110 = r741103 - r741109;
        return r741110;
}
↓
double f(double F, double l) {
        double r741111 = atan2(1.0, 0.0);
        double r741112 = l;
        double r741113 = r741111 * r741112;
        double r741114 = tan(r741113);
        double r741115 = F;
        double r741116 = r741114 / r741115;
        double r741117 = 1.0;
        double r741118 = r741117 / r741115;
        double r741119 = r741116 * r741118;
        double r741120 = r741113 - r741119;
        return r741120;
}

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 8.4

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

  2. Simplified8.0

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

  4. Applied *-un-lft-identity8.0

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

  5. Applied times-frac0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))