Average Error: 16.4 → 12.3
Time: 8.4s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\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 (* PI l)) F)))))
double code(double F, double l) {
	return ((double) (((double) (((double) M_PI) * l)) - ((double) ((1.0 / ((double) (F * F))) * ((double) tan(((double) (((double) M_PI) * l))))))));
}
↓
double code(double F, double l) {
	return ((double) (((double) (((double) M_PI) * l)) - (1.0 / (F / (((double) tan(((double) (((double) M_PI) * l)))) / 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 clear-num_binary6416.2

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

  5. Simplified12.3

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

  6. Final simplification12.3

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

Reproduce

herbie shell --seed 2020210 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))