Average Error: 16.8 → 12.6
Time: 9.2s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell + 1 \cdot \frac{-1}{F \cdot \frac{F}{\tan \left(\pi \cdot \ell\right)}}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell + 1 \cdot \frac{-1}{F \cdot \frac{F}{\tan \left(\pi \cdot \ell\right)}}
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)) + ((double) (1.0 * (-1.0 / ((double) (F * (F / ((double) tan(((double) (((double) M_PI) * l))))))))))));
}

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 Error: 16.8 bits

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

  2. SimplifiedError: 16.5 bits

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

  4. Applied clear-numError: 16.5 bits

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

  5. SimplifiedError: 12.6 bits

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

  6. Final simplificationError: 12.6 bits

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

Reproduce

herbie shell --seed 2020200 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))