Average Error: 16.3 → 12.3
Time: 15.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\right)
double f(double F, double l) {
        double r64 = atan2(1.0, 0.0);
        double r65 = l;
        double r66 = r64 * r65;
        double r67 = 1.0;
        double r68 = F;
        double r69 = r68 * r68;
        double r70 = r67 / r69;
        double r71 = tan(r66);
        double r72 = r70 * r71;
        double r73 = r66 - r72;
        return r73;
}
↓
double f(double F, double l) {
        double r74 = atan2(1.0, 0.0);
        double r75 = l;
        double r76 = r74 * r75;
        double r77 = 1.0;
        double r78 = F;
        double r79 = r77 / r78;
        double r80 = 1.0;
        double r81 = tan(r76);
        double r82 = r78 / r81;
        double r83 = r77 / r82;
        double r84 = r80 * r83;
        double r85 = r79 * r84;
        double r86 = r76 - r85;
        return r86;
}

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 16.3

    \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity16.3

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

  4. Applied times-frac16.3

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

  5. Applied associate-*l*12.3

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

  7. Applied div-inv12.3

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

  8. Applied associate-*l*12.3

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

  9. Simplified12.2

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

  11. Applied clear-num12.3

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

  12. Final simplification12.3

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))