Average Error: 16.8 → 12.4
Time: 9.0s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}}{F}
(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l)
 :precision binary64
 (- (* PI l) (/ (/ (sin (* PI l)) (* F (cos (* PI l)))) 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) - ((sin(((double) M_PI) * l) / (F * cos(((double) M_PI) * l))) / 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 16.8

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

  2. Simplified16.6

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

  4. Applied clear-num_binary6416.6

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

  5. Simplified12.4

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

  7. Applied *-un-lft-identity_binary6412.4

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

  8. Applied *-un-lft-identity_binary6412.4

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

  9. Applied times-frac_binary6412.4

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

  10. Applied *-un-lft-identity_binary6412.4

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

  11. Applied times-frac_binary6412.4

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

  12. Applied add-sqr-sqrt_binary6412.4

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

  13. Applied times-frac_binary6412.4

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

  14. Simplified12.4

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

  15. Simplified12.4

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

  17. Applied tan-quot_binary6412.4

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

  18. Applied associate-/l/_binary6412.4

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

  19. Final simplification12.4

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

Alternatives

Reproduce

herbie shell --seed 2021106 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))