Average Error: 16.4 → 0.7
Time: 7.4s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \log \left({\left(\sqrt[3]{e^{\pi \cdot \ell}}\right)}^{F}\right)}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \log \left({\left(\sqrt[3]{e^{\pi \cdot \ell}}\right)}^{F}\right)}}{F}
(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l)
 :precision binary64
 (-
  (* PI l)
  (/ (/ 1.0 (- (/ F (* PI l)) (log (pow (cbrt (exp (* PI l))) F)))) 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 / (((double) M_PI) * l)) - log(pow(cbrt(exp(((double) M_PI) * l)), F)))) / 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 16.4

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

  2. Simplified16.2

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

  4. Applied associate-/r*_binary6412.3

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

  6. Applied clear-num_binary6412.3

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

  7. Taylor expanded around 0 8.3

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\color{blue}{\frac{F}{\pi \cdot \ell} - 0.3333333333333333 \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}}}{F}\]

  8. Simplified8.3

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\color{blue}{\frac{F}{\pi \cdot \ell} - 0.3333333333333333 \cdot \left(\left(\pi \cdot \ell\right) \cdot F\right)}}}{F}\]
  9. Using strategy rm

  10. Applied add-log-exp_binary6412.9

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \color{blue}{\log \left(e^{0.3333333333333333 \cdot \left(\left(\pi \cdot \ell\right) \cdot F\right)}\right)}}}{F}\]

  11. Simplified0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2020224 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))