Average Error: 17.1 → 12.8
Time: 8.7s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
↓
\[\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F} \]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l) :precision binary64 (- (* PI l) (/ (/ (tan (* 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) - ((tan(((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 17.1

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

  2. Simplified16.9

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

  4. Applied clear-num_binary6416.9

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

  5. Simplified12.8

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

  7. Applied *-un-lft-identity_binary6412.8

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

  8. Applied *-un-lft-identity_binary6412.8

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

  9. Applied times-frac_binary6412.8

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

  10. Applied *-un-lft-identity_binary6412.8

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

  11. Applied times-frac_binary6412.8

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

  12. Applied add-cube-cbrt_binary6412.8

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

  13. Applied times-frac_binary6412.8

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

  14. Simplified12.8

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

  15. Simplified12.8

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

  17. Applied *-un-lft-identity_binary6412.8

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

  18. Final simplification12.8

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

Reproduce

herbie shell --seed 2021198 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))