Average Error: 16.8 → 12.4
Time: 6.5s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{\frac{F}{1}}{\tan \left(\pi \cdot \ell\right)}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{\frac{\frac{F}{1}}{\tan \left(\pi \cdot \ell\right)}}}{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) - ((1.0 / ((F / 1.0) / tan((((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. Using strategy rm

  3. Applied associate-/r*16.8

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

  4. Applied associate-*l/12.4

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

  6. Applied clear-num12.4

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

  7. Applied associate-*l/12.4

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

  8. Simplified12.4

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

  10. Applied clear-num12.4

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

  11. Final simplification12.4

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))