Average Error: 16.9 → 12.5
Time: 8.5s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\sqrt{\sqrt{\pi}} \cdot \left(\ell \cdot {\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)}^{1.5}\right)\right)}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\sqrt{\sqrt{\pi}} \cdot \left(\ell \cdot {\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)}^{1.5}\right)\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)
   (/
    (tan
     (*
      (sqrt (sqrt PI))
      (* l (pow (* (sqrt (sqrt PI)) (sqrt (sqrt PI))) 1.5))))
    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) * (tan(sqrt(sqrt((double) M_PI)) * (l * pow((sqrt(sqrt((double) M_PI)) * sqrt(sqrt((double) M_PI))), 1.5))) / 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 16.9

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

  2. Simplified16.6

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

  4. Applied *-un-lft-identity_binary6416.6

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

  5. Applied times-frac_binary6412.5

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

  7. Applied add-sqr-sqrt_binary6412.7

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

  8. Applied associate-*l*_binary6412.7

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

  9. Simplified12.7

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

  11. Applied add-sqr-sqrt_binary6412.6

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

  12. Applied associate-*l*_binary6412.6

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

  13. Simplified12.7

    \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\sqrt{\sqrt{\pi}} \cdot \color{blue}{\left(\ell \cdot {\left(\sqrt{\pi}\right)}^{1.5}\right)}\right)}{F}\]
  14. Using strategy rm

  15. Applied add-sqr-sqrt_binary6412.5

    \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\sqrt{\sqrt{\pi}} \cdot \left(\ell \cdot {\color{blue}{\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)}}^{1.5}\right)\right)}{F}\]

  16. Final simplification12.5

    \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\sqrt{\sqrt{\pi}} \cdot \left(\ell \cdot {\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)}^{1.5}\right)\right)}{F}\]

Reproduce

herbie shell --seed 2020233 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))