Average Error: 16.1 → 8.3
Time: 8.1s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell + 1 \cdot \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell} + \pi \cdot \left(\ell \cdot \left(F \cdot -0.3333333333333333\right)\right)}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell + 1 \cdot \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell} + \pi \cdot \left(\ell \cdot \left(F \cdot -0.3333333333333333\right)\right)}
(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l)
 :precision binary64
 (+
  (* PI l)
  (*
   1.0
   (/ (/ -1.0 F) (+ (/ F (* PI l)) (* PI (* l (* F -0.3333333333333333))))))))
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 / F) / ((double) ((F / ((double) (((double) M_PI) * l))) + ((double) (((double) M_PI) * ((double) (l * ((double) (F * -0.3333333333333333)))))))))))));
}

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: 16.1 bits

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

  2. SimplifiedError: 15.8 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: 15.9 bits

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

  5. SimplifiedError: 11.9 bits

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

  7. Applied associate-/r*Error: 11.9 bits

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

  8. Taylor expanded around 0 Error: 8.3 bits

    \[\leadsto \pi \cdot \ell - 1 \cdot \frac{\frac{1}{F}}{\color{blue}{\frac{F}{\pi \cdot \ell} - 0.3333333333333333 \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}}\]

  9. SimplifiedError: 8.3 bits

    \[\leadsto \pi \cdot \ell - 1 \cdot \frac{\frac{1}{F}}{\color{blue}{\frac{F}{\pi \cdot \ell} + \pi \cdot \left(\ell \cdot \left(F \cdot -0.3333333333333333\right)\right)}}\]

  10. Final simplificationError: 8.3 bits

    \[\leadsto \pi \cdot \ell + 1 \cdot \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell} + \pi \cdot \left(\ell \cdot \left(F \cdot -0.3333333333333333\right)\right)}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))