Average Error: 16.3 → 12.2
Time: 8.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \frac{1}{\frac{F}{1 \cdot \tan \left(\pi \cdot \ell\right)}}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \frac{1}{\frac{F}{1 \cdot \tan \left(\pi \cdot \ell\right)}}
double f(double F, double l) {
        double r15942 = atan2(1.0, 0.0);
        double r15943 = l;
        double r15944 = r15942 * r15943;
        double r15945 = 1.0;
        double r15946 = F;
        double r15947 = r15946 * r15946;
        double r15948 = r15945 / r15947;
        double r15949 = tan(r15944);
        double r15950 = r15948 * r15949;
        double r15951 = r15944 - r15950;
        return r15951;
}
↓
double f(double F, double l) {
        double r15952 = atan2(1.0, 0.0);
        double r15953 = l;
        double r15954 = r15952 * r15953;
        double r15955 = 1.0;
        double r15956 = F;
        double r15957 = r15955 / r15956;
        double r15958 = 1.0;
        double r15959 = tan(r15954);
        double r15960 = r15958 * r15959;
        double r15961 = r15956 / r15960;
        double r15962 = r15955 / r15961;
        double r15963 = r15957 * r15962;
        double r15964 = r15954 - r15963;
        return r15964;
}

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 associate-*l/12.2

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

  9. Applied clear-num12.2

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

  10. Final simplification12.2

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))