Average Error: 16.6 → 12.1
Time: 8.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\sqrt{1} \cdot \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\sqrt{1} \cdot \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\right)
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) - ((sqrt(1.0) / F) * (sqrt(1.0) * (1.0 / (F / tan((((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 16.6

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

  3. Applied add-sqr-sqrt16.6

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

  4. Applied times-frac16.6

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

  5. Applied associate-*l*12.1

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

  7. Applied div-inv12.1

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

  8. Applied associate-*l*12.1

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

  9. Simplified12.1

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

  11. Applied clear-num12.1

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

  12. Final simplification12.1

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))