Average Error: 16.4 → 12.5
Time: 12.1s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\left(\sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right)\]
\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) (((double) (((double) M_PI) * l)) - ((double) (((double) (1.0 / ((double) (F * F)))) * ((double) tan(((double) (((double) M_PI) * l))))))));
}
↓
double code(double F, double l) {
	return ((double) (((double) (((double) M_PI) * l)) - ((double) (((double) (1.0 / F)) * ((double) (((double) (((double) cbrt(((double) (((double) (1.0 / F)) * ((double) tan(((double) (((double) M_PI) * l)))))))) * ((double) cbrt(((double) (((double) (1.0 / F)) * ((double) tan(((double) (((double) M_PI) * l)))))))))) * ((double) cbrt(((double) (((double) (1.0 / F)) * ((double) tan(((double) (((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 16.4

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

  3. Applied *-un-lft-identity16.4

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

  4. Applied times-frac16.4

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

  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 add-cube-cbrt12.5

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

  8. Final simplification12.5

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

Reproduce

herbie shell --seed 2020162 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))