Average Error: 17.1 → 12.8
Time: 13.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r16060 = atan2(1.0, 0.0);
        double r16061 = l;
        double r16062 = r16060 * r16061;
        double r16063 = 1.0;
        double r16064 = F;
        double r16065 = r16064 * r16064;
        double r16066 = r16063 / r16065;
        double r16067 = tan(r16062);
        double r16068 = r16066 * r16067;
        double r16069 = r16062 - r16068;
        return r16069;
}
↓
double f(double F, double l) {
        double r16070 = atan2(1.0, 0.0);
        double r16071 = l;
        double r16072 = r16070 * r16071;
        double r16073 = 1.0;
        double r16074 = F;
        double r16075 = r16073 / r16074;
        double r16076 = 1.0;
        double r16077 = r16076 / r16074;
        double r16078 = sqrt(r16070);
        double r16079 = r16078 * r16071;
        double r16080 = r16078 * r16079;
        double r16081 = tan(r16080);
        double r16082 = r16077 * r16081;
        double r16083 = r16075 * r16082;
        double r16084 = r16072 - r16083;
        return r16084;
}

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 17.1

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

  3. Applied *-un-lft-identity17.1

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

  4. Applied times-frac17.1

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

  5. Applied associate-*l*12.7

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

  7. Applied add-sqr-sqrt12.8

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

  8. Applied associate-*l*12.8

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

  9. Final simplification12.8

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))