Average Error: 16.7 → 12.4
Time: 9.4s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1 \cdot \frac{\tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{F}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1 \cdot \frac{\tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{F}}{F}
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) (1.0 * ((double) (((double) tan(((double) (((double) sqrt(((double) M_PI))) * ((double) (((double) sqrt(((double) M_PI))) * l)))))) / F)))) / F))));
}

Error

Bits error versus F

Bits error versus l

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.7

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

  3. Applied *-un-lft-identity16.7

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

  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*12.3

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

  7. Applied div-inv12.3

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

  8. Applied associate-*l*12.3

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

  9. Simplified12.3

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

  11. Applied associate-*r/12.3

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

  12. Applied associate-*r/12.3

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

  13. Simplified12.3

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

  15. Applied add-sqr-sqrt12.4

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

  16. Applied associate-*l*12.4

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

  17. Final simplification12.4

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

Reproduce

herbie shell --seed 2020174 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))