Average Error: 16.5 → 12.5
Time: 9.4s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{\sqrt{1}}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt{\sqrt{1}}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{\sqrt{1}}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt{\sqrt{1}}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\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) - ((sqrt(1.0) / F) * ((sqrt(sqrt(1.0)) / (cbrt(F) * cbrt(F))) * ((sqrt(sqrt(1.0)) / cbrt(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 add-sqr-sqrt16.5

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

  4. Applied times-frac16.5

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

  5. Applied associate-*l*12.3

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

  7. Applied add-cube-cbrt12.5

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

  8. Applied add-sqr-sqrt12.5

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

  9. Applied sqrt-prod12.5

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

  10. Applied times-frac12.5

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

  11. Applied associate-*l*12.5

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

  12. Final simplification12.5

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))