Average Error: 16.6 → 12.6
Time: 7.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1 \cdot \frac{\frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F} \cdot \sqrt[3]{F}}}{\sqrt[3]{F}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1 \cdot \frac{\frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F} \cdot \sqrt[3]{F}}}{\sqrt[3]{F}}}{F}
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 * ((tan((((double) M_PI) * l)) / (cbrt(F) * cbrt(F))) / cbrt(F))) / F));
}

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 *-un-lft-identity16.6

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

  4. Applied times-frac16.7

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

  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 div-inv12.4

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

  8. Applied associate-*l*12.4

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

  9. Simplified12.4

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

  11. Applied associate-*r/12.4

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

  12. Applied associate-*r/12.4

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

  13. Simplified12.4

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

  15. Applied add-cube-cbrt12.6

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

  16. Applied associate-/r*12.6

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

  17. Final simplification12.6

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))