Average Error: 16.8 → 12.6
Time: 9.5s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\tan \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}\right)\right)\right) \cdot \frac{1}{F}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\tan \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}\right)\right)\right) \cdot \frac{1}{F}\right)
(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l)
 :precision binary64
 (-
  (* PI l)
  (*
   (/ 1.0 F)
   (*
    (tan
     (*
      (* PI (* (cbrt l) (cbrt l)))
      (* (cbrt (cbrt l)) (* (cbrt (cbrt l)) (cbrt (cbrt l))))))
    (/ 1.0 F)))))
double code(double F, double l) {
	return ((double) (((double) (((double) M_PI) * l)) - ((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) ((1.0 / F) * ((double) (((double) tan(((double) (((double) (((double) M_PI) * ((double) (((double) cbrt(l)) * ((double) cbrt(l)))))) * ((double) (((double) cbrt(((double) cbrt(l)))) * ((double) (((double) cbrt(((double) cbrt(l)))) * ((double) cbrt(((double) cbrt(l)))))))))))) * (1.0 / 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 Error: 16.8 bits

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

  3. Applied *-un-lft-identityError: 16.8 bits

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

  4. Applied times-fracError: 16.8 bits

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

  5. Applied associate-*l*Error: 12.3 bits

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

  6. SimplifiedError: 12.3 bits

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

  8. Applied add-cube-cbrtError: 12.5 bits

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

  9. Applied associate-*r*Error: 12.5 bits

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

  11. Applied add-cube-cbrtError: 12.6 bits

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

  12. Final simplificationError: 12.6 bits

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

Reproduce

herbie shell --seed 2020205 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))