Average Error: 16.5 → 12.6
Time: 9.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\left(\sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\left(\sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right)
double code(double F, double l) {
	return ((((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l))));
}
↓
double code(double F, double l) {
	return ((((double) M_PI) * l) - ((1.0 / F) * ((cbrt(((1.0 / F) * tan((((double) M_PI) * l)))) * cbrt(((1.0 / F) * tan((((double) M_PI) * l))))) * cbrt(((1.0 / F) * tan((((double) M_PI) * l)))))));
}

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 16.5

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

  3. Applied *-un-lft-identity16.5

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

  4. Applied times-frac16.6

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

  5. Applied associate-*l*12.4

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

  7. Applied add-cube-cbrt12.6

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

  8. Final simplification12.6

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))