Average Error: 8.3 → 0.6
Time: 42.5s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}}{F}\]
double f(double F, double l) {
        double r622944 = atan2(1.0, 0.0);
        double r622945 = l;
        double r622946 = r622944 * r622945;
        double r622947 = 1.0;
        double r622948 = F;
        double r622949 = r622948 * r622948;
        double r622950 = r622947 / r622949;
        double r622951 = tan(r622946);
        double r622952 = r622950 * r622951;
        double r622953 = r622946 - r622952;
        return r622953;
}
↓
double f(double F, double l) {
        double r622954 = atan2(1.0, 0.0);
        double r622955 = l;
        double r622956 = r622954 * r622955;
        double r622957 = 1.0;
        double r622958 = F;
        double r622959 = tan(r622956);
        double r622960 = r622958 / r622959;
        double r622961 = r622957 / r622960;
        double r622962 = r622961 / r622958;
        double r622963 = r622956 - r622962;
        return r622963;
}


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

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 8.3

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

  2. Simplified7.9

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

  4. Applied associate-/r*0.6

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

  6. Applied *-un-lft-identity0.6

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

  7. Applied associate-/l*0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))