Average Error: 8.4 → 0.7
Time: 39.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F}}{F}\]
\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) \cdot \frac{1}{F}}{F}
double f(double F, double l) {
        double r617419 = atan2(1.0, 0.0);
        double r617420 = l;
        double r617421 = r617419 * r617420;
        double r617422 = 1.0;
        double r617423 = F;
        double r617424 = r617423 * r617423;
        double r617425 = r617422 / r617424;
        double r617426 = tan(r617421);
        double r617427 = r617425 * r617426;
        double r617428 = r617421 - r617427;
        return r617428;
}
↓
double f(double F, double l) {
        double r617429 = atan2(1.0, 0.0);
        double r617430 = l;
        double r617431 = r617429 * r617430;
        double r617432 = tan(r617431);
        double r617433 = 1.0;
        double r617434 = F;
        double r617435 = r617433 / r617434;
        double r617436 = r617432 * r617435;
        double r617437 = r617436 / r617434;
        double r617438 = r617431 - r617437;
        return r617438;
}

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 associate-/r*0.7

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

  6. Applied div-inv0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))