Average Error: 8.3 → 0.6
Time: 40.5s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}\]
double f(double F, double l) {
        double r578023 = atan2(1.0, 0.0);
        double r578024 = l;
        double r578025 = r578023 * r578024;
        double r578026 = 1.0;
        double r578027 = F;
        double r578028 = r578027 * r578027;
        double r578029 = r578026 / r578028;
        double r578030 = tan(r578025);
        double r578031 = r578029 * r578030;
        double r578032 = r578025 - r578031;
        return r578032;
}
↓
double f(double F, double l) {
        double r578033 = atan2(1.0, 0.0);
        double r578034 = l;
        double r578035 = r578033 * r578034;
        double r578036 = tan(r578035);
        double r578037 = F;
        double r578038 = r578036 / r578037;
        double r578039 = 1.0;
        double r578040 = r578039 / r578037;
        double r578041 = r578038 * r578040;
        double r578042 = r578035 - r578041;
        return r578042;
}


\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}

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 *-un-lft-identity7.9

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

  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 2019102 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))