Average Error: 16.6 → 8.5
Time: 9.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1}}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1}}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}
double f(double F, double l) {
        double r18983 = atan2(1.0, 0.0);
        double r18984 = l;
        double r18985 = r18983 * r18984;
        double r18986 = 1.0;
        double r18987 = F;
        double r18988 = r18987 * r18987;
        double r18989 = r18986 / r18988;
        double r18990 = tan(r18985);
        double r18991 = r18989 * r18990;
        double r18992 = r18985 - r18991;
        return r18992;
}
↓
double f(double F, double l) {
        double r18993 = atan2(1.0, 0.0);
        double r18994 = l;
        double r18995 = r18993 * r18994;
        double r18996 = 1.0;
        double r18997 = sqrt(r18996);
        double r18998 = F;
        double r18999 = r18997 / r18998;
        double r19000 = r18998 / r18995;
        double r19001 = 0.3333333333333333;
        double r19002 = r18998 * r18995;
        double r19003 = r19001 * r19002;
        double r19004 = r19000 - r19003;
        double r19005 = r18997 / r19004;
        double r19006 = r18999 * r19005;
        double r19007 = r18995 - r19006;
        return r19007;
}

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 16.6

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

  3. Applied add-sqr-sqrt16.6

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

  4. Applied times-frac16.7

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

  5. Applied associate-*l*12.6

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

  7. Applied associate-*l/12.6

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

  9. Applied associate-/l*12.6

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

  10. Taylor expanded around 0 8.5

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

  11. Final simplification8.5

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

Reproduce

herbie shell --seed 2020060 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))