Average Error: 17.1 → 8.5
Time: 7.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}\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) (((double) (1.0 / F)) * ((double) (1.0 * ((double) (1.0 / ((double) (((double) (F / ((double) (((double) M_PI) * l)))) - ((double) (0.3333333333333333 * ((double) (F * ((double) (((double) M_PI) * l))))))))))))))));
}

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 17.1

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

  3. Applied *-un-lft-identity17.1

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

  4. Applied times-frac17.1

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

  5. Applied associate-*l*12.7

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

  7. Applied div-inv12.7

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

  8. Applied associate-*l*12.7

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

  9. Simplified12.7

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

  11. Applied clear-num12.7

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

  12. Taylor expanded around 0 8.5

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

  13. Final simplification8.5

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

Reproduce

herbie shell --seed 2020122 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))