Average Error: 15.8 → 0.7
Time: 25.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right) \cdot \frac{1}{3}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right) \cdot \frac{1}{3}}}{F}
double f(double F, double l) {
        double r614807 = atan2(1.0, 0.0);
        double r614808 = l;
        double r614809 = r614807 * r614808;
        double r614810 = 1.0;
        double r614811 = F;
        double r614812 = r614811 * r614811;
        double r614813 = r614810 / r614812;
        double r614814 = tan(r614809);
        double r614815 = r614813 * r614814;
        double r614816 = r614809 - r614815;
        return r614816;
}
↓
double f(double F, double l) {
        double r614817 = atan2(1.0, 0.0);
        double r614818 = l;
        double r614819 = r614817 * r614818;
        double r614820 = 1.0;
        double r614821 = F;
        double r614822 = r614821 / r614819;
        double r614823 = exp(r614819);
        double r614824 = log(r614823);
        double r614825 = r614821 * r614824;
        double r614826 = 0.3333333333333333;
        double r614827 = r614825 * r614826;
        double r614828 = r614822 - r614827;
        double r614829 = r614820 / r614828;
        double r614830 = r614829 / r614821;
        double r614831 = r614819 - r614830;
        return r614831;
}

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 15.8

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

  2. Simplified12.1

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

  4. Applied clear-num12.2

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

  5. Taylor expanded around 0 8.1

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\color{blue}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}}}{F}\]
  6. Using strategy rm

  7. Applied add-log-exp0.7

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \color{blue}{\log \left(e^{\pi \cdot \ell}\right)}\right)}}{F}\]

  8. Final simplification0.7

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right) \cdot \frac{1}{3}}}{F}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))