Average Error: 16.9 → 12.8
Time: 9.1s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\left(\sqrt[3]{\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{F}}}\right) \cdot \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\left(\sqrt[3]{\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{F}}}\right) \cdot \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\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) (((double) cbrt(((double) (1.0 / F)))) * ((double) cbrt(((double) (1.0 / F)))))) * ((double) (((double) (((double) cbrt(((double) (1.0 / ((double) (((double) cbrt(F)) * ((double) cbrt(F)))))))) * ((double) cbrt(((double) (1.0 / ((double) cbrt(F)))))))) * ((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 16.9

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

  3. Applied *-un-lft-identity16.9

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

  4. Applied times-frac17.0

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

  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-cube-cbrt12.8

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

  8. Applied associate-*l*12.8

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

  10. Applied add-cube-cbrt12.8

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

  11. Applied cbrt-prod12.8

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

  12. Simplified12.8

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

  13. Simplified12.8

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

  14. Final simplification12.8

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

Reproduce

herbie shell --seed 2020153 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))