Average Error: 8.6 → 0.7
Time: 32.6s
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 r283226 = atan2(1.0, 0.0);
        double r283227 = l;
        double r283228 = r283226 * r283227;
        double r283229 = 1.0;
        double r283230 = F;
        double r283231 = r283230 * r283230;
        double r283232 = r283229 / r283231;
        double r283233 = tan(r283228);
        double r283234 = r283232 * r283233;
        double r283235 = r283228 - r283234;
        return r283235;
}
↓
double f(double F, double l) {
        double r283236 = atan2(1.0, 0.0);
        double r283237 = l;
        double r283238 = r283236 * r283237;
        double r283239 = tan(r283238);
        double r283240 = F;
        double r283241 = r283239 / r283240;
        double r283242 = 1.0;
        double r283243 = r283242 / r283240;
        double r283244 = r283241 * r283243;
        double r283245 = r283238 - r283244;
        return r283245;
}

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 8.6

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

  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-frac0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))