Average Error: 17.3 → 0.8
Time: 8.4s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell + 1 \cdot \left(\frac{1}{F} \cdot \frac{-1}{F \cdot \log \left({\left({\left(e^{\pi}\right)}^{\ell}\right)}^{\frac{-1}{3}}\right) + \frac{F}{\pi \cdot \ell}}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell + 1 \cdot \left(\frac{1}{F} \cdot \frac{-1}{F \cdot \log \left({\left({\left(e^{\pi}\right)}^{\ell}\right)}^{\frac{-1}{3}}\right) + \frac{F}{\pi \cdot \ell}}\right)
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) (1.0 * ((double) (((double) (1.0 / F)) * ((double) (-1.0 / ((double) (((double) (F * ((double) log(((double) pow(((double) pow(((double) exp(((double) M_PI))), l)), -0.3333333333333333)))))) + ((double) (F / ((double) (((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 17.3

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

  2. Simplified17.0

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

  4. Applied clear-num17.0

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

  5. Simplified12.8

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

  6. Taylor expanded around 0 8.7

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

  7. Simplified8.7

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

  9. Applied add-sqr-sqrt8.7

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

  10. Applied times-frac8.7

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

  11. Simplified8.7

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

  12. Simplified8.7

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

  14. Applied add-log-exp0.8

    \[\leadsto \pi \cdot \ell - 1 \cdot \left(\frac{1}{F} \cdot \frac{1}{F \cdot \color{blue}{\log \left(e^{\left(\pi \cdot \ell\right) \cdot \frac{-1}{3}}\right)} + \frac{F}{\pi \cdot \ell}}\right)\]

  15. Simplified0.8

    \[\leadsto \pi \cdot \ell - 1 \cdot \left(\frac{1}{F} \cdot \frac{1}{F \cdot \log \color{blue}{\left({\left({\left(e^{\pi}\right)}^{\ell}\right)}^{\frac{-1}{3}}\right)} + \frac{F}{\pi \cdot \ell}}\right)\]

  16. Final simplification0.8

    \[\leadsto \pi \cdot \ell + 1 \cdot \left(\frac{1}{F} \cdot \frac{-1}{F \cdot \log \left({\left({\left(e^{\pi}\right)}^{\ell}\right)}^{\frac{-1}{3}}\right) + \frac{F}{\pi \cdot \ell}}\right)\]

Reproduce

herbie shell --seed 2020185 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))