Average Error: 16.4 → 12.5
Time: 7.8s
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 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 add-cube-cbrt_binary6416.3

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

  5. Applied associate-/l*_binary6416.3

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

  6. Simplified14.9

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

  8. Applied frac-2neg_binary6414.9

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

  9. Applied distribute-frac-neg_binary6414.9

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

  10. Simplified12.5

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

  12. Applied frac-2neg_binary6412.5

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

  13. Applied distribute-frac-neg_binary6412.5

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

  14. Final simplification12.5

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

Reproduce

herbie shell --seed 2021211 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))