Average Error: 16.4 → 12.4
Time: 8.4s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\ell \cdot \sqrt{\pi}\right)\right)}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\ell \cdot \sqrt{\pi}\right)\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)
   (/ (* (sin (* PI l)) (/ 1.0 F)) (cos (* (sqrt PI) (* l (sqrt PI))))))))
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) sin(((double) (((double) M_PI) * l)))) * (1.0 / F))) / ((double) cos(((double) (((double) sqrt(((double) M_PI))) * ((double) (l * ((double) sqrt(((double) M_PI))))))))))))));
}

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-identity_binary6416.4

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

  4. Applied times-frac_binary6416.5

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

  5. Applied associate-*l*_binary6412.4

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

  6. Simplified12.4

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

  8. Applied add-sqr-sqrt_binary6412.4

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

  9. Applied associate-*l*_binary6412.4

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

  10. Simplified12.4

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

  12. Applied tan-quot_binary6412.4

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

  13. Applied associate-*l/_binary6412.4

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

  14. Simplified12.4

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

  15. Final simplification12.4

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

Reproduce

herbie shell --seed 2020210 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))